A mutation-selection model for general genotypes with recombination

We investigate a continuous time, probability measure-valued dynamical system that describes the process of mutation-selection balance in a context where the population is infinite, there may be infinitely many loci, and there are weak assumptions on selective costs. Our model arises when we incorporate very general recombination mechanisms into a previous model of mutation and selection from Steinsaltz, Evans and Wachter (2005) and take the relative strength of mutation and selection to be sufficiently small. The resulting dynamical system is a flow of measures on the space of loci. Each such measure is the intensity measure of a Poisson random measure on the space of loci: the points of a realization of the random measure record the set of loci at which the genotype of a uniformly chosen individual differs from a reference wild type due to an accumulation of ancestral mutations. Our motivation for working in such a general setting is to provide a basis for understanding mutation-driven changes in age-specific demographic schedules that arise from the complex interaction of many genes, and hence to develop a framework for understanding the evolution of aging. We establish the existence and uniqueness of the dynamical system, provide conditions for the existence and stability of equilibrium states, and prove that our continuous-time dynamical system is the limit of a sequence of discrete-time infinite population mutation-selection-recombination models in the standard asymptotic regime where selection and mutation are weak relative to recombination and both scale at the same infinitesimal rate in the limit.


Abstract
We investigate a continuous time, probability measure-valued dynamical system that describes the process of mutation-selection balance in a context where the population is infinite, there may be infinitely many loci, and there are weak assumptions on selective costs. Our model arises when we incorporate very general recombination mechanisms into a previous model of mutation and selection from Steinsaltz, Evans and Wachter (2005) and take the relative strength of mutation and selection to be sufficiently small. The resulting dynamical system is a flow of measures on the space of loci. Each such measure is the intensity measure of a Poisson random measure on the space of loci: the points of a realization of the random measure record the set of loci at which the genotype of a uniformly chosen individual differs from a reference wild type due to an accumulation of ancestral mutations. Our motivation for working in such a general setting is to provide a basis for understanding mutation-driven changes in age-specific demographic schedules that arise from the complex interaction of many genes, and hence to develop a framework for understanding the evolution of aging.
We establish the existence and uniqueness of the dynamical system, provide conditions for the existence and stability of equilibrium states, and prove that our continuous-time dynamical system is the limit of a sequence of discrete-time infinite population mutation-selection-recombination models in the standard asymptotic regime where selection and mutation are weak relative to recombination and both scale at the same infinitesimal rate in the limit.

Introduction
A principal goal of population genetics is to understand how the mechanisms of mutation, selection and recombination determine the manner in which the distribution of genotypes in a population changes over time. The distribution of genotypes governs, to a large degree, the distribution of phenotypes, an object that is both more readily observable and of more immediate practical interest. Conversely, the differential survival of genotypes is mediated almost exclusively by their phenotypic expression.
Mathematical models have particular salience for the study of senescence. Progressive physical deterioration and mortality rates rising across the lifespan are not, on the face of it, characteristics that natural selection would be expected to favor. Peter Medawar, in his inaugural lecture at University College London delivered in 1951 [Med52] called the origin of senescence "an unsolved problem in biology." He went on to outline a possible solution. His idea, called "mutation accumulation" presents senescence as a side-effect of the interplay between natural selection and mutation. Shaping evolutionary thinking for the last half-century, Medawar's idea is the centerpiece of the present work.
Medawar's proposal was complemented in 1957 by ideas of George C. Williams [Wil57] directing attention to alleles with multiple, that is, pleiotropic, effects on fitness, some enhancing and some impairing the production and survival of progeny. This concept of "antagonistic pleiotropy" has taken its place beside the concept of "mutation accumulation"; together they are the twin guiding themes for evolutionary theories of aging. The relevance of both concepts to senescence turns on the truism that deleterious effects that only take hold later in life sacrifice less of an organism's remaining potential for reproduction and nurturing than effects that act early. Medawar leads us to think of an equilibrium between mutation and selection in which a population is maintained at less than optimal fitness. Williams depicts alleles that could in constant environments go on to fixation.
It is important to emphasize that the mutations under consideration here are germ-line mutations passed on from generation to generation over stretches of evolutionary time. Mutations may also occur within cells of a single individual during the individual's lifespan and be passed on and spread through cell division. These "somatic mutations" may also be important for aging. Although the models in [SEW05] might turn out to have some relevance to somatic mutation, the focus in this work is on germ-line mutations.
There is a rich literature on models for mutation, selection, and recombination, described further in Section 1.5. Our models build on that tradition but differ from much of it in being specially designed to accommodate general forms of detailed demographic structure. In leading applications, mutant alleles are associated with representations of age-specific effects on demographic schedules. One goal is to 1. INTRODUCTION provide structure from which predictions of the shapes of curves of probabilities of death as functions of age can be derived from assumptions about rates and kinds of mutations. This enterprise has come to the fore with the discovery of common features in the age-specific demographic schedules of populations from a variety of species which seem to call out for evolutionary explanations and which have stimulated the field of biodemography ([WF97] [CT03]).
An early precedent for the kinds of models we develop here is found in the wellknown work of Motoo Kimura and Takeo Maruyama [KM66]. Our demographic formulations grow out of the work of Brian Charlesworth [Cha94], [Cha01].
Recapitulating, mutation accumulation models that incorporate the countervailing forces of recurring, slightly deleterious mutations and persistent selection have been staples of evolutionary theory [B00]. In order to use these models to explain phenomena such as aging that are presumed to result from the combination of many small mutational impacts, it is necessary to adopt a multilocus perspective. However, as Pletcher and Curtsinger [PC98] point out, early progress in this area relied on simplifying assumptions that are severely limiting and possibly unfounded: equal impacts from all mutations, additive effects of mutations on age-specific survival, and the existence of mutations that impact a specific, narrow range of ages.
Several general multilocus formalisms have been proposed. Notable among these is that of [KJB02], which is designed to allow almost any conceivable regime of selection, mating, linkage, mutation, and phenotypic effects. Such frameworks have not been exploited for studies of aging, in part because so much detail is counterproductive when what concerns us is not the fate of any individual allele but rather the mass of overlapping, age-varying phenotypic effects that are central to standard theories of aging.
Kevin Dawson [Daw99] applied a variant of the Kimura-Maruyama rare-allele approximation [KM66,B00] (see also [Kon82]) to aging. While this less detailed view of the genome is more amenable to theoretical analysis, it is also not suited to describing the interacting phenotypic contribution of multiple loci. In a previous paper [SEW05], we proposed a model, which we also describe briefly here in the development leading to (1.5), that overcame many of these limitations. That model leads to computable solutions for mutation-selection equilibria, the hazard functions which such equilibria imply, and the time evolution of the population distribution of genotypes.
In this work, we define and analyze a parallel model that, in essence, incorporates recombination into the model of [SEW05]. The key assumptions behind the model in this work are the following.
• The population is infinite.
• The genome may consist of infinitely many or even a continuum of loci.
• Reproduction is sexual, in that each individual has two parents.
• The mechanism of genetic recombination randomly shuffles together the genomes of the parents in order to obtain the genome of the offspring. • Mating is random and individuals are identical except for their genotypes, so the population at any time can be completely described by a probability measure on the space of possible genotypes -heuristically, the empirical distribution of genotypes in the population or, equivalently, the probability distribution of the genotype of a randomly sampled individual.
• Individuals are haploid -an individual has only one copy of each gene rather than copies from each of two parents. • Down a lineage, mutant alleles only accumulate -there is no backmutation to cancel out the alleles introduced by earlier mutation events. • Fitness is calculated for individuals rather than for mating pairs. • A genotype becomes less fit when it accumulates additional mutant alleles, but otherwise selective costs may be freely specified. • Recombination acts on a faster time scale than mutation or selectionin other words, the common quasi-linkage equilibrium (QLE) assumption holds.
Our general picture of the genome and the processes of mutation, selection and recombination is similar to that of [BT91,KJB02]. Whereas [BT91,KJB02] invoke the QLE assumption to justify treating the effects of alleles at different loci as nearly independent, we present a detailed asymptotic treatment in a standard, explicit scaling regime.
More specifically, we establish conditions under which a discrete-time dynamical system with the above features converges to a continuous-time, deterministic dynamical system that has as its state space the probability measures on the set of possible genotypes. At any point in time, the genotype distribution is in the complete linkage equilibrium represented by the distribution of a Poisson random measure. The points of a realization of the Poisson random measure represent loci at which a randomly sampled individual's genotype has accumulated ancestral mutations away from an original "reference" genotype which we call the wild type. Because a Poisson random measure is defined by its intensity measure, which in our case is a finite measure on the set of loci, the asymptotic model can be more simply described by a deterministic dynamical system that moves about in the space of such measures.
In order to establish such a convergence result it is, of course, not enough simply to show that the distribution of the genotype of a randomly chosen individual at a fixed time converges to the distribution of a Poisson random measure. Rather, we must keep track of the accumulated perturbations that arise over time from the effect of mutation, selection and recombination, and demonstrate the convergence of the entire time evolution of the genotype distribution to a dynamical system of Poisson probability measures.
Moreover, we establish conditions under which the continuous-time dynamical system has equilibria, investigate when the system converges to an equilibrium from the pure wild type genotype, and obtain results about the stability and attractivity of that particular equilibrium when it is present.
Our models provide a basis for the rigorous study of a number of questions in the biodemography of longevity, as surveyed in [Wac03]: • the adequacy of Charlesworth's [Cha01] proposed explanation of Gompertz hazards and mortality plateaus as a consequence of mutation accumulation, • the causes of the hyperexponential hazards described by Horiuchi [Hor03], • extensions to age-structured settings of Haldane's principle that, in its original form, equates the mutation rate to population decline of fitness, • contrasts between proportional and additive mutation effects on hazards as discussed in [Bau05,Bau08], • the possibility of mortality rates diverging to infinity at ages before the end of reproduction, which have been termed "Walls of Death". We pursue these matters in companion papers [WES08,WSE10].

Informal description of the limit model
In this section we describe the asymptotic model and explain by way of motivation why it is reasonable that such a dynamical system should arise as the limit of a sequence of discrete-time systems with the features listed above. We present a formal development of the model in Sections 2.1 and 2.2 of Chapter 2. The exposition in [CE09] was based on an early version of the present text.
Denote by M the collection of loci in the portion of the genome that is of interest to us. There is a distinguished reference wild type genotype, and each locus represents a "position" at which the genotype of an individual may differ from that of the wild genotype. We allow the set M to be quite general and do not necessarily think of it as a finite collection of physical DNA base positions or a finite collection of genes. For example, the proposed explanation for the Gompertz mortality curve and mortality plateaus at extreme ages in Charlesworth [Cha01] suggest taking M to be a class of functions from R + to R + : the value of such a function at age x ≥ 0 represents an additional increment to mortality at age x conferred by a mutation away from the wild type at this locus. Some structure on M is necessary to accommodate rigorous probability theory, so we take M to be a complete, separable metric space.
The genotype of an individual is specified by the set of loci at which there has been a mutation somewhere along the ancestral lineage leading to that individual. More precisely, a genotype is an element of the space G of integer-valued finite Borel measures on M. An element of G is a finite sum i δ mi , where δ m is the unit point mass at the locus m ∈ M. The measure i δ mi corresponds to a genotype that has ancestral mutations at loci m 1 , m 2 , . . .. The wild type genotype is thus the null measure. We do not require that the loci m i ∈ M be distinct. We thus allow several copies of a mutation. This is reasonable, since we do not identify mutations with changes in nucleotide sequences in a one-to-one manner.
For example, if M is finite, so we might as well take M = {1, 2, . . . , N } for some positive integer N , then G is essentially the Cartesian product N N 0 of N copies of the nonnegative integers. A genotype is of the form N j=1 n j δ j , indicating that an ancestral mutation is present n j times at locus j, and we identify such a genotype with the nonnegative integer vector (n 1 , n 2 , . . . , n N ).
Recall that the population is infinite and all that matters about an individual is the individual's genotype, so that the dynamics of the population are described by the proportions of individuals with genotypes that belong to the various subsets of G. We are thus led to consider a family of probability measures P t , t ≥ 0, on G, where P t (G) for some subset G ⊆ G represents the proportion of individuals in the population at time t that have genotypes belonging to G. Note that we may also think of P t (G) as the probability that an individual chosen uniformly at random from the population has a genotype belonging to the set G. In other words, P t is the distribution of a random finite integer-valued measure on M. For example, if M = {1, 2, . . . , N } and we identify G with the Cartesian product N N 0 as above, then P t ({(n 1 , n 2 , . . . , n N )}) represents the probability that an individual chosen uniformly at random from the population has n j ancestral mutations at locus j for j = 1, 2, . . . , N .
We next indicate how we model mutation, selection and recombination to obtain the evolution dynamics for P t , t ≥ 0.
Mutation alone. Suppose that there is only mutation and no selection or recombination. In this case all individuals present in the population at a given time die and reproduce at the same rate, because differing genotypes do not confer differing selective costs. Mutations accumulate down lineages because they cannot be replaced by recombination.
We describe the mutation process using a finite measure ν on the space of loci M, where ν(B) for B ⊆ M gives the rate at which mutations from the ancestral wild type belonging to the set B accumulate along a given lineage.
Write P t [Φ] = G Φ(g) dP t (g) for some bounded Borel test function Φ : G → R. That is, P t [Φ] is the expected value of the real-valued random variable obtained by applying the function Φ to the genotype of an individual chosen uniformly at random from the population. The content of our assumptions when there is only mutation is that P t , t ≥ 0, evolves according to where e j is the j th coordinate vector. This equation is, of course, just a special case of the usual equation (see, for example, Section III.1.2 of [B00]) describing evolution due to mutation of type frequencies in a population where the set of types is N N and mutation from type n to type n + e j occurs at rate ν({j}). The evolution equation (1.1) is of the form d dt P t = AP t for a certain linear operator A. We recognize that A is the infinitesimal generator of a G-valued Lévy process, and hence (1.1) has the following explicit probabilistic solution: LetΠ denote a Poisson random measure on M × R + with intensity measure ν ⊗λ, where ⊗ denotes the product of measures andλ is Lebesgue measure. That is,Π is a random integer-valued Borel measure such that: (1) The nonnegative integer-valued random variableΠ(A) is Poisson with expectation (ν ⊗λ)(A) for any Borel subset A of M × R + . (2) If A 1 , A 2 , . . . , A n are disjoint Borel subsets of M × R + , then the random variablesΠ(A k ) are independent.
Define a G-valued random variable Z t (that is, Z t is a random finite integer-valued measure on M) by δ m dΠ((m, u)). Then, where W is a random measure on M that has probability measure P 0 and is independent ofΠ. In particular, if P 0 is itself the distribution of a Poisson random measure, then P t is also the distribution of a Poisson random measure. If we write ρ t for the intensity measure associated with P t (that is, ρ t is the measure on M defined by ρ t (A) := G g(A) dP t (g) for A ⊆ M), then ρ t evolves according to the simple dynamics ρ t (A) = ρ 0 (A) + t ν(A).
Selection alone. Now suppose there is only selection and no mutation or recombination. We specify the fitnesses of different genotypes by a selective cost function S : G → R + . The difference S(g ) − S(g ) for g , g ∈ G is the difference in the rate of sub-population growth between the sub-population of individuals with genotype g and the sub-population of individuals with genotype g . We make the normalizing assumption S(0) = 0 and suppose that in line with our assumption above that genotypes with more accumulated mutations are less fit. It follows that at time t ≥ 0 the per individual rate of increase of the proportion of the population of individuals with genotype g is P t [S] − S(g ). More formally, S(g ) dP t (g ) dP t (g ). If S is non-epistatic, that is, if S has the additive property then the selective effects of different mutations do not interact. In particular, if P 0 is the distribution of a Poisson random measure, then P t is also the distribution of a Poisson random measure and, writing ρ t for the intensity measure associated with P t as before, we have ρ t (dm ) = ρ 0 (dm ) − However, in the more general case in which S is epistatic (that is, non-additive), then P t is, in general, not the distribution of a Poisson random measure -even when P 0 is.
Combining mutation and selection. If there is mutation and selection, but no recombination, then the appropriate evolution equation for P t comes from simply combining equations (1.1) and (1.4): (1.5) This is the model introduced and analyzed at length in [SEW05] using the Feynman-Kac formula. When M = {1, 2, . . . , N }, (1.5) is a special case of the classical system of ordinary differential equations for mutation and selection in continuous-time. See Section III.1.2 of [B00] for the derivation of an analytic solution that agrees with the one that arises from a Feynman-Kac analysis. The principal result from [SEW05] is the following: Proposition 1.1. Let Z t be the random measure distributed according to the Poisson probability measureΠ in (1.2). Suppose that there is a positive T such that for all t ∈ [0, T ). Then, (1.5) has a solution on [0, T ), given by Notice that the effect of selection may be understood as reweighting a Poisson measure, rather than explicitly removing mutations from the population. This will be an important theme in the convergence proof that starts in Chapter 4.
The model without recombination in [SEW05], like the model presented here, is an infinite-population model. Alleles are not subject to genetic drift. The fittest genotypes present in the initial population are not lost, no matter how rare they are. A finite-population model without recombination defined along the same lines would be vulnerable to Mueller's ratchet, the process in which the fittest classes of genotypes in an asexual population can be successively lost through drift, carrying the population to extinction. A thorough discussion of the ratchet and the associated advantages of sex and recombination with references is found in [B00], pages 303-308. In the face of Mueller's ratchet, a finite-population version of the model in [SEW05] would not be viable. This observation underscores the importance of incorporating recombination, which avoids the ratchet by allowing fittest classes to be reconstituted in every generation.
Recombination alone. The effect of recombination is to choose an individual uniformly at random from the population at some rate and replace the individual's genotype g with a genotype of the form g (· ∩ M ) + g (· ∩ M ), where g is the genotype of another randomly chosen individual, M is a subset of M chosen according to a suitable random mechanism, and M is the complement of M . Thus, recombination randomly shuffles together two different genotypes drawn from the population. In order to specify the recombination mechanism fully, we would need to specify the recombination rate and the distribution of the segregating set M . Suppose, however, that we are in the following regime.
• Recombination acts alone; that is, there is no mutation or selection.
• The initial population P 0 has the property that there does not exist an m ∈ M with P 0 ({g ∈ G : g({m}) > 0}) > 0; that is, no single mutation from the ancestral wild type is possessed by a positive proportion of the initial population (but see Remark 1.2 and Section 5.4 below). • The mechanism for choosing the segregating set M is such that, loosely speaking, if m and m are two loci, then there is positive probability that m ∈ M and m ∈ M ; that is, no region of the genome M is immune from the shuffling effect of recombination. Then, under these conditions, the probability measure P t converges as t → ∞ to the distribution of a Poisson random measure on M with the same intensity measure as P 0 . Moreover, the speed of this convergence increases with the recombination rate and so if we take the recombination rate to be effectively infinite, then the probability measure P t is essentially Poisson for all t > 0 with the same intensity measure as P 0 , irrespective of the details of the recombination mechanism.
Combining mutation, selection and recombination. We have seen that if P 0 is the distribution of a Poisson random measure, then mutation preserves this property. On the other hand, epistatic selection drives the population distribution away from Poisson, while increasing rates of recombination push it towards Poisson. Thus, when all three processes operate and we consider a limiting regime where recombination acts on a much faster time scale than selection and recombination, we expect asymptotically that P t is also the distribution of a Poisson random measure for all t > 0. As before, ρ t denotes the intensity measure of P t (so that ρ t is a finite measure on the space M of loci). In anticipation of Notation 2.1, we write X π for a Poisson random measure on M with intensity measure π. Combining our previous observations, we expect that ρ t should satisfy the evolution equation We define the rigorous counterpart of (1.8) in Chapter 2 and establish the existence and uniqueness of solutions. Furthermore, we show in Chapters 4 ff. that our dynamical equation is indeed a limit of a sequence of standard discrete generation, mutation-selection-recombination models.
Given that it involves computing an expected value for a quite general Poisson random measure, (1.8) may look rather forbidding. However, for certain reasonable choices of selective costs we can evaluate the integral explicitly, leading to a simpler and more intuitive system. We consider three such cases in the following three sections. First, though, we make a useful observation about how we may clump loci together in (1.8).
Remark 1.2. Consider two instances (ρ t ) t≥0 and (ρ t ) t≥0 of the dynamical system (1.8) with respective locus spaces M and M , associated genotype spaces G and G , mutation intensity measures ν and ν , and selective costs S and S . Suppose that there is a Borel measurable map T from M onto M such that the following hold.
• The initial measure ρ 0 on M is the push-forward of the initial measure ρ 0 on M by the map T . • The mutation intensity measure ν on M is the push-forward of the mutation intensity measure ν on M by the map T . • The selective cost S on G has the property that S (g ) = S (h ) whenever the push-forwards of g , h ∈ G by T are the same. • The selective cost S (g ) on G is given by the common value of S (g ) for all g ∈ G that have push-forward by T equal to g . Then, for each t > 0, the measure ρ t on M is the push-forward of the measure ρ t on M by the map T . Intuitively, we have a situation in which for a given g ∈ G any two genotypes in the set T −1 (g ) ⊆ G are indistinguishable in terms of their associated selective cost, and so we may identify any two such genotypes as being the same. Of course, we cannot recover the finer description ρ t from the coarser one ρ t in general, but if we write P t and P t for the population distributions of genotypes corresponding to ρ t and ρ t (that is, P t and P t are the distributions of Poisson random measures on M and M with intensity measures ρ t and ρ t ), then the push-forward of P t by S is the same as the push-forward of P t by S . That is, the population distributions of selective costs agree whether we use the fine or the coarse description of genotypes.

Example I: Mutation counting
The simplest special case of our framework occurs when there are many loci but the selective cost of a genotype g only depends on the total number of loci g(M) at which there have been ancestral mutations away from the wild type.
For example, suppose that the space M of loci is the unit interval [0, 1] and the selective cost S is of the form S(g) = s(g(M)) for some non-decreasing function s : N 0 → R + with s(0) = 0.
We may apply Remark 1.2 and "replace" the locus space M by a single point. Let q := ν(M) be the total rate at which mutations occur and write r t := ρ t (M) for the expected number of ancestral mutations in the genotype of an individual chosen at random from the population at time t ≥ 0. It follows from Remark 1.2 that the function r evolves autonomously according to the (non-linear) ordinary differential equationṙ One example of this simplified model, which lines up with models familiar from earlier literature, assumes the cost per mutation to be a constants, so that s(k) =sk and ψ(x) =sx, and assumes the starting point to be the null genotype. In this case, we readily compute that the intensity at time t is r t = (q/s) 1 − exp{−st} . The intensity converges monotonically to the equilibrium value q/s, the elementary expression for mutation-selection equilibrium going back to J.B.S. Haldane. Thus, the limiting distribution for the number of mutations from the ancestral wild type is Poisson with mean q/s. Our assumption about recombination leads to different, simpler answers than Kimura and Maruyama's treatment of mutation counting without recombination (see [KM66]).
For another example, consider the case when costs are multiplicative, in the sense that s(k) := 1 − exp{−kθ} for some constant θ > 0. We have ψ(x) = ax exp{−ax}, where a := 1 − exp{−θ}. An equilibrium exists only if the mutation rate q is below the maximum of r exp{−r}, namely 1/e, and in that case r t converges monotonically to the smallest positive solution of the equation q − ψ(x) = 0. The solution is x = (1/a) ∞ n=1 (−n) n−1 (−q) n / n! where the summed expression is Lambert's W function evaluated at the negative argument −q. The power series expansion is a simple application of the technique known variously as reversion of series or the Lagrange inversion formula. These properties for multiplicative costs are generalized in Section 3.2. Note also that when θ is small the equilibrium is approximately q/θ, as one would expect from the observation that in this case s(k) is approximately θk for small k and hence this model is approximately the additive one of the previous paragraph withs = θ.
If we are interested only in how the population distribution of selective costs evolves, then we need consider only (r t ) t≥0 , rather than (ρ t ) t≥0 . However, we should be somewhat careful about how we interpret the biological import of this simplification. We justified the dynamical system (1.8) as describing the evolving population distribution of genotypes defined in terms of the locus space M in a population undergoing mutation, selection and recombination. Mathematically, we see that in this example we can replace the locus space [0, 1] by a single point for the purposes of studying the dynamics of the distribution of selective costs, but this does not mean that biologically the multilocus model is identical with a single locus model. As we show later, our instance of (1.8) with locus space [0, 1] arises as a limit of discrete generation models in which Poisson random measures appear because of the manner in which recombination breaks up and shuffles together genotypes from different individuals. Even though our instance of (1.8) with locus space a single point is mathematically well-defined, it cannot arise as a limit of such discrete generation models because with a single locus there is no way that recombination can drive genotype distributions towards Poisson. Single-point mutation spaces require the special treatment described in Remark 1.2.

Example II: Polynomial selective costs
Recall that when M = {1, 2, . . . , N } we can encode genotypes as ordered N −tuples of nonnegative integers, where the entry in the k th coordinate is the number of ancestral mutations present at locus k. Then, (1.8) becomes the system of ordinary differential equations As in the previous example, we should not think of {1, 2, . . . , N } as being the "real" locus space. Rather, we should imagine that there is something like a continuum of loci which are partitioned into N sub-classes in such a way that the loci in each sub-class have indistinguishable selective effects, and (1.9) is a reduced description that comes from applying the observation in Remark 1.2.
A natural family of selective costs is given by those of the polynomial form where the sum is taken over all nonempty subsets I ⊆ {1, . . . , N } and we adopt the usual multi-index convention that for a vector v the notation v I denotes the product i∈I v i . The constant α {i} for 1 ≤ i ≤ N measures the selective cost of mutation i alone, whereas the constant α I for a subset I ⊆ {1, . . . , N } of cardinality greater than one measures the selective cost attributable to interactions between all of the mutations in I over and above that attributable to interactions between mutations in proper subsets of I.
The system of ordinary differential equations (1.9) becomes where we write ρ k := ρ({k}) and ν k := ν({k}), and where I k denotes the collection of subsets of {1, . . . , N } that contain k, see [CE09]. It is shown in [CE09] that if ν k > 0 for all k i (that is, if mutations may occur at all loci), if α {i} > 0 for all i (that is, if the individual effect of any mutation is deleterious, in keeping with our general assumption on the selective cost), and if α I ≥ 0 for all subsets I (that is, the synergistic effects of individually deleterious mutations are never beneficial), then the system of equations (1.10) has a unique equilibrium point in the positive orthant R N + . Moreover, this equilibrium is globally attractive; that is, the system converges to the equilibrium from any initial conditions in R N + . The condition α I ≥ 0 for all I certainly implies our standing assumption that S(g + h) ≥ S(g) for all g, h ∈ G, but it is strictly stronger. It is also shown in [CE09] that the analogue of this result for general M holds with a suitable definition of polynomial selective costs in terms of sums of integrals against products of the measure g with itself.

Example III: Demographic selective costs
The following model is discussed in [WES08,WSE10], where there is a more extensive discussion of the demographic assumptions.
For the moment, suppose that the space of loci M is general. Write x (g) for the probability that an individual with genotype g ∈ G lives beyond age x ∈ R + . At age x, the corresponding cumulative hazard is − log x (g) and the hazard function is its derivative with respect to x when it exists. . Suppose that the infinitesimal rate that an individual at age x ∈ R + has offspring is f x , independently of the individual's genotype. For individuals with genotype g, the size of the next generation relative to the current one is ∞ 0 f x x (g) dx, the "Net Reproduction Ratio" or "NRR". Suppose further that there is a background hazard λ and that an ancestral mutation at locus m ∈ M contributes a bounded increment θ(m, x) to the cumulative hazard function at age x. This function θ with two arguments is a generalization of the constant cost θ in Section 1.2 and of a single-argument cost θ(m) in play in examples with multiplicative costs in Section 3.2.
The probability that an individual with genotype g ∈ G lives beyond age x ∈ R + is (1.11) x (g) = exp −λx − M θ(m, x) dg(m) .

INTRODUCTION
The corresponding selective cost is (Recall that selective costs represent relative rates of increase, and we have adopted the normalizing convention that S(0) = 0). Marginal selective costs are given by the expression Expected marginal costs have the expected value of x (g) under the integral on the right-hand side. When the genotype g is a realization of a Poisson random measure X π whose intensity is the finite measure π on M, this expected survivorship function can be found from the Poisson identity The identity applies to bounded Borel functions φ : M → R. It is proved, for instance, in [Kin93,3.15]. It also follows from Campbell's Theorem (see Proposition A.8).
The identity gives the expression, in terms of the expectation operator E, An important issue for demographic applications is whether the solution ρ t , t ≥ 0, of (1.8) converges to an equilibrium ρ * as t → ∞ and, if so, what are the features of that equilibrium. In particular, what we can say about E[ x (X ρ * )], the probability that a randomly chosen individual from the equilibrium population lives beyond age x? It is not hard to show that if the limit ρ * exists, then it must be absolutely continuous with respect to the mutation rate measure ν and have a Radon-Nikodym derivative r * that satisfies We study such equilibria further in Section 3.9 and in [WES08,WSE10]. In particular, in [WES08], we consider mutations which each provide a point mass increment to the hazard at a specific age. When M = R + and the mutation rate measure ν is absolutely continuous with respect to Lebesgue measure, and θ is of the form θ(m, x) = η(m)1 {x≥m} , (where, as before, 1 A is the indicator function of the set A) the equilibrium equation turns out to be equivalent to a second-order, nonlinear, ordinary differential equation in one variable that can be solved explicitly for r * .

Comments on the literature
We make some brief remarks about the substantial literature on multilocus deterministic models in population genetics and its relation to our work.
A very comprehensive reference is Reinhard Bürger's book [B00] along with Bürger's review paper [Bür98]). As well as giving an overview of the classical models for finitely many alleles at each of a finite number of loci, these works consider at length deterministic haploid continuum-of-alleles models in which individuals have a type that is envisioned as the contribution of a gene to a given quantitative trait. The type belongs to a general state space that represents something like the trait value (in which case the state space is a subset of R) and is often regarded as the combined effect of a multilocus genotype. Each type has an associated fitness, which is some fairly arbitrary function from the type space to (0, 1]. However, the models do not explicitly incorporate a family of loci, the configuration of alleles present at those loci, or a function describing the fitness of a configuration. Rather, everything is cast in terms of how fit each type is and how likely one type is to mutate into another.
Certain classes of mutation-selection models without recombination are solved explicitly in [WBG98,BW01] using ideas from statistical mechanics. Such models may be treated either as multilocus systems with complete linkage or as structured single locus systems. We also mention the constellation of papers [BB03,Baa05,Baa07,Baa01] presenting a deterministic model of population change due to recombination alone. In our setting, the Poisson probability measures constitute, very roughly speaking, a hypersurface within the space of all probability measures and recombination can be viewed as an operation that pushes an arbitrary probability measure "towards" that hypersurface. General results about the convergence of discrete-time dynamical systems to a continuous-time one that is forced onto a submanifold by a suitable vector field may be relevant here, but we are not aware of a particular framework of this kind into which we can fit our results.
Finally, we make two comments about our model in order to distinguish it from others in the literature that superficially might seem to have similar features.
• We do not present a Markovian stochastic model such as the one in [SH92], where the population is described in terms of an evolving Poisson random measure that keeps track of the proportion of alleles at each site that are mutant (with a typical site being purely wild type and only exceptional sites having a positive proportion of mutants present). For us, the evolution of the proportions of different genotypes in the population is described by a deterministic dynamical system living on a space of probability measures. If we sample from the evolving probability measure at some fixed time, then the resulting individual's genotype is a Poisson random measure. • Linkage arises in our discrete-time approximation models as the dependence between loci, a natural consequence of non-additive selection costs, is only partially broken up in any finite number of rounds of recombination. However, linkage does not appear in the limit model. That is, if we sample from the limit population at some time, then the fact that the resulting individual's genotype is described by a Poisson random measure means that the presence of ancestral mutations from wild type in one part of the genotype is independent of the presence of mutations in another part. Our convergence theorem in Chapter 8 therefore delineates the relative strengths of mutation, selection and recombination that lead asymptotically to a situation in which the Poissonizing effect of recombination wins out over the interactions introduced by non-additive selection. Understanding how these two forces counteract each other is far from trivial and is the most demanding technical task of the present work. We cannot stress too strongly that we have not a priori assumed that linkage is absent.
1.6. Overview of the remainder of the work We define the measure-valued dynamical system (1.8) rigorously in Chapter 2 and establish the existence and uniqueness of solutions.
We investigate in Chapter 3 whether the dynamical system has equilibria and whether these equilibria are stable and attractive.
We devote Chapters 4 to 8 to showing that the dynamical system is a limit of discrete-generation, infinite-population models. We define the discrete-generation models in Chapter 4 and we preview the convergence theorem and its hypotheses in Chapter 5. As a first step towards the proof of the convergence result, we show in Chapter 6 that an analogous convergence result holds when the recombination mechanism is replaced by a complete Poissonization operation that destroys all linkage between loci in a single step. The proof of the actual convergence result is quite involved and requires a number of technical estimates that, loosely speaking, bound the extent to which selection reintroduces linkage that has been partially removed by recombination. We present these preliminary results in Chapter 7. We state the convergence theorem and complete its proof in Chapter 8.
The appendix contains relevant versions of results from the literature as well as technical results about Poisson random measures and Radon-Nikodym derivatives that are used throughout the main text. for g, h ∈ G. An element of G represents a "genotype" regarded as the set of loci at which there have been ancestral mutations away from the reference wild type. The null measure represents the wild genotype.
Recall also that the state of the population at time t ≥ 0 in our model is a probability measure P t on G that is the distribution of a Poisson random measure on M. The distribution of such a Poisson random measure is determined by its intensity measure, which is in general a locally-finite Borel measure on M and which is a finite measure when the mutation measure ν has finite total mass. For most results, we do assume that ν has finite total mass, but a brief discussion of infinite-mass cases is given in Section 2.6.
• Denote by H the Banach space of finite signed Borel measures on M equipped with the norm · Was defined below. • Let H + be the subset of H consisting of nonnegative measures. • For π ∈ H, write π + , π − ∈ H + for the positive and negative parts of π appearing in the Hahn-Jordan decomposition. Thus, π = π + − π − . • For any measure P on G, let µP be the intensity measure associated with P ; that is, for a nonnegative Borel function f on M, • For π ∈ H + , write X π for a Poisson random measure on M with intensity measure π and Π π for the distribution of this Poisson random measure.
The norms on our spaces of measures are based on a metric from the class of metrics named for Leonid Wasserstein (also transliterated as Vasershtein, or otherwise). In our investigations we have to deal with spaces of measures at a hierarchy of different levels, including G, H, and finite signed measures on G. The Wasserstein metrics provide a unified way of topologizing all these various spaces.
• Given a metric space (E, d), let Lip be the space of functions f : E → R such that • Define a norm · Was on the space of finite signed Borel measures on (E, d) by The Wasserstein metric metrizes the topology of weak convegence of measures on the finite signed Borel measures on (E, d).
• For any measure π in such a space of signed measures, write for the (possibly infinite) Lipschitz constant of the selective cost function S.
We take advantage of the versatility of these definitions to build Wasserstein metrics on top of Wasserstein metrics. We start by taking M for the metric space E. Then G, as a space of measures on M, has a Wasserstein metric. Next we take G with its Wasserstein metric for the metric space E, and obtain a Wasserstein metric on the finite signed measures on G, including the measures P t . Also, with M again playing the role of E, we obtain a Wasserstein metric on H.
An extensive account of Wasserstein metrics may be found in [Rac91,RR98]. The properties used here are described in Problem 3.11.2 of [EK86]. In particular, it is shown that our version is indeed a metric and that H as well as G and the finite signed measures on G are complete in this norm. The Wasserstein distance between two probability measures is bounded above by their total variation distance.
We note that the designation "Wasserstein metric" is also often applied to the analogous definition where the constraining Lipschitz norm · Lip does not include the supremum norm term. This latter distance is always greater than or equal to the Wasserstein metric as we are defining it. It is equivalent to the Kantorovich-Rubinstein distance, which is a member of the class of Monge-Kantorovich distances, a class defined by a single parameter p; the Kantorovich-Rubinstein distance is the element of this class corresponding to p = 1. Details may be found in [Vil03, Section 7.1], [Vil09, Chapter 6] or [AGS05, Section 7.1].
In some places we need to use the total variation norm, defined for a finite signed Borel measure π as (2.4) π TV := sup f dπ : sup There is a corresponding Lipschitz constant for a function F : G → R, defined as This yields the identity Lastly, we sometimes compare two equivalent probability measures P and Q on G by considering the quantity Note that Θ(P, P ) = 0. Conversely, if Θ(P, Q) = 0, then log dQ/dP is a constant and so dQ/dP is also a constant; however, this constant must be 1 since P and Q are probability measures, and hence P = Q. Note also that Θ(P, Q) = Θ(Q, P ) because log dP/dQ = log(dQ/dP ) −1 = − log dQ/dP . Finally, if R is another probability measure equivalent to P and Q, then and so Θ(P, R) ≤ Θ(P, Q) + Θ(Q, R). Therefore, Θ restricted to a suitable class of probability measures is a metric.

Definition of the dynamical system
The state of the population at time t ≥ 0 is given by a probability measure P t on G that is the distribution of a Poisson random measure with intensity measure ρ t . Our informal description of the evolution of ρ t , and hence of P t , is motivated by (1.8), restated here.
In order to formalize this definition, it is convenient to introduce the following two objects.
for x ∈ M and π ∈ H + . • Define the operator D : That is, for any bounded Borel function f : M → R and π in H + , With this notation, our evolution equation (2.8) becomes (2.10) Dρ s ds A solution is an H + -valued function ρ that is continuous with respect to the Wasserstein metric and therefore with respect to the topology of weak convergence of measures and satisfies (2.10) for all t ≥ 0. Equation (2.10) involves the integration of a measure-valued function, and such an integral can have a number of different meanings. We require only that if η : R + → H is a Borel function, then for t ≥ 0 the integral t 0 η s ds is the element of H satisfying, for every Borel A ⊆ M,

Existence and uniqueness of solutions
We now prove the existence and uniqueness of solutions to (2.10) with the assistance of three lemmas that are proved below in Section 2.4. The proof is an application of the standard iterative method of CharlesÉmile Picard. Dα + s ds.
Note that ∆ maps Γ into itself. Moreover, for α, β ∈ Γ, By Lemma 2.7 below, the norm inside the integral on the right has the following bound We conclude that Thus, ∆ : Γ → Γ is a contraction, provided c is chosen sufficiently large. It follows from the Contraction Mapping Theorem that the equation has a unique solution in Γ. Furthermore, any function in H that is a solution to (2.12) must automatically be in Γ. Therefore, the solution is unique. It remains to show that it actually takes values in the subset H + of nonnegative measures. For any Borel set A ⊆ M, In particular, t → ρ t (A) is continuous. For any Borel set A ⊂ M we have . The Lipschitz condition on S in terms of the Wasserstein metric implies, via Lemma 2.7 below, that F π (m) ≤ σ for all m ∈ M and π ∈ H + . Then, Hence, ρ t (A) ≥ 0 for 0 ≤ t ≤ 1/σ. Because this holds for all Borel sets A ⊆ M, we have ρ t ∈ H + for 0 ≤ t ≤ 1/σ. Iterating this argument, with the time 0 replaced successively by the times 1/σ, 2/σ, . . . gives the result.

Lemmas used in the proof of existence and uniqueness
We assume in this section that the hypotheses of Theorem 2.4 hold and prove the lemmas on which that theorem relies.
Lemma 2.5. The function F π is Lipschitz in the Wasserstein metric for each finite measure π ∈ H + , and, in terms of the Lipschitz constant σ of the selective cost function S, sup and sup Furthermore, for all m ∈ M and all π ∈ H + , Proof. By definition, implying the bound on 1/F π .
Lemma 2.7. For two finite signed measures α, β ∈ H, Proof. Suppose without loss of generality that α + (M) ≤ β + (M). By Lemmas 2.5 and 2.6, for any Lipschitz test function f , We have used the fact that f f Lip ≤ f Lip f Lip for f , f ∈ Lip.

Density form of the dynamical system
The solutions provided by Theorem 2.4 are absolutely continuous with respect to ρ 0 + ν and so can be written out in terms of Radon-Nikodym derivatives (that is, densities) with respect to that reference measure. We show below that, more generally, the Radon-Nikodym derivatives with respect to suitable reference measures ζ belong to the space L ∞ + (M, ζ) of nonnegative functions that are essentially-bounded for ζ. This parallel approach of viewing our dynamical system as taking values in a space of functions rather than a space of measures was first developed in [CE09]. It provides another route to defining the informal dynamical system rigorously and establishing the existence and uniqueness of solutions.
Of course our analysis as a whole cannot be carried through in L ∞ (M, ζ). The singleton mutations δ m in G with which selective costs are measured can not be viewed as elements of L ∞ (M, ζ) for some measure ζ equivalent to ρ 0 + ν unless ρ 0 or ν has atoms, and the simultaneous consideration of a continuum of nonequivalent initial measures ρ 0 can not be accommodated within a single L ∞ (M, ζ). For the full theory, spaces of measures equipped with Wasserstein metrics or tools of similar generality are required. However, proofs of existence and uniqueness for fixed initial conditions and mutation measures can be carried out in the L ∞ (M, ζ) setting. Moreover, this approach facilitates an extension of the theory to some cases where the mutation measure ν has infinite total mass and it also justifies a step in the proof of Lemma 3.14 in our chapter on equilibria.
• The measure ζ on M is any measure with respect to which the initial state ρ 0 and the mutation measure ν are both absolutely continuous, with bounded Radon-Nikodym derivatives r 0 (m) and q ν (m). (In practice, it is convenient to take ζ to be the sum of ν and the part of ρ 0 , if any, orthogonal to ν). Instead of working forward from conditions on the selective cost S via the lemmas of Section 2.4, we impose a bound and Lipschitz condition directly on the whole contribution from Dρ. Anticipating the remarks in Section 2.6, we offer proofs that do not rely on the finiteness of ν(M).
Theorem 2.9. Fix a mutation measure ν and starting state ρ 0 in H + and a selective cost S : G → R + such that S(0) = 0, and S(g) ≤ S(g + h) for all g, h ∈ G. Let ν and ρ 0 be absolutely continuous with respect to a measure ζ in H + with bounded Radon-Nikodym derivatives q ν (m) and r 0 (m).
Suppose the function (π, m) → F π (m) defined from S in Definition 2.3 the following conditions.
• There is a constant σ such that for all π ∈ H + and m ∈ M, F π (m) ≤ σ.
Proof. The proof mimics the proof of Theorem 2.4. The space H is replaced by K, H + is replaced by K + , and the norm · Was is replaced by · ∞ . The analogue of the subset Γ consists of K-valued functions α with α 0 = ρ 0 and The assumptions imposed on F take the place of the bounds from Lemmas 2.7 and 2.5. Theorem 2.9 gives us for each t ≥ 0 a measure which, as an element of K + , has a Radon-Nikodym derivative with respect to ζ. A priori, the Radon-Nikodym derivative is defined separately for each t ≥ 0 as a function of m ∈ M and is only unique up to ζ-null sets. The behavior of a particular choice for the ensemble of Radon-Nikodym derivatives is, for fixed m ∈ M, not guaranteed to be even a measurable function of t ≥ 0. We now show that we can define versions of the Radon-Nikodym derivatives that are in fact continuously differentiable functions of t ≥ 0 for every m ∈ M.
Theorem 2.10. Suppose that ρ t is a solution to the dynamic equation (2.10) in C(R + , K + ) for selective costs satisfying the conditions of Theorem 2.9. Then, there exists a Borel function (t, m) → r t (m) on R + × M such that for all t ≥ 0 the function m → r t (m) is a Radon-Nikodym derivative of ρ t with respect to ζ and such that for all m ∈ M the function t → r t (m) is continuously differentiable with a derivative which satisfieṡ Proof. We will construct a curve of a priori new measures ξ t with Radon-Nikodym derivatives m → x t (m) that have nice behavior in t ≥ 0 for fixed m ∈ M. We then prove that, as measures, ξ = ρ. Finally, we replace the arbitrary choice of Radon-Nikodym derivative r t for ρ t whose existence is guaranteed by Theorem 2.9 with x t .
Because t → ρ t is continuous, it follows from a monotone class argument that . is a suitable increasing sequence of finitely generated sub-σ-fields. Suppose for k ∈ N that F k consists of the empty set and unions of collections of sets drawn from the Borel partition {A k,1 , . . . , A k,n(k) } of M. Then, we may suppose that if the limit exists, 0, otherwise (see, for example, Section III-1 of [Nev75]). In particular, we may suppose that the map (t, m) → r t (m) is Borel measurable. When ρ t is already known, for each fixed m ∈ M the function F ρt (m) is just some known Borel function of t and the Radon-Nikodym derivative m → r 0 (m) is just some known bounded function. Consider, then, for each fixed m ∈ M the following ordinary integral equation for an unknown function x t (m), analogous to our dynamic equation For every m ∈ M, (2.14) has the continuously differentiable solution (2.15) For t ≥ 0, define measures ξ t by ξ t (dm) = x t (m)ζ(dm). For ξ t , for any Borel set A, (2.16) For ρ t , with Radon-Nikodym derivatives r t , we have a similar equation but with the integral over m ∈ M inside the integral over s ∈ R + , namely We want to reverse the order of integration in the first equation for ξ. We can apply Fubini's Theorem so long as the function (s, m) → x s (m) ∈ R is Borel measurable, which is the case (because of the equation defining x t (m)) when (s, m) → F ρs (m) is Borel measurable. As we have already remarked, the map s → ρ s is continuous by construction, and the map (π, m) → F π (m) is always Borel measurable. (Under the conditions of Theorem 2.4 it is jointly continuous.) Thus, we can apply Fubini's Theorem.
After reversing the order of integration in (2.16), we subtract (2.17) from (2.16): The terms in the initial state and the mutation measure cancel Thus, For Borel sets A with strictly positive measure, and particularly for sets A n , n ∈ N, defined by we can divide both sides of (2.18) by ζ(A). The resulting left-hand side is no less than β t − 1/n for all n ∈ N. Taking the supremum over n ∈ N, we conclude for all Gronwall's Inequality (see Appendix A.1) then forces β t ≡ 0. It follows that x t (m) = r t (m) outside of a set of m ∈ M with ζ measure zero so that, for every t, x t (m) is itself a Radon-Nikodym derivative of ρ t with respect to ζ. In other words, ξ t = ρ t and ξ t satisfies the dynamic equation. We replace our original To spare notation, we henceforth write r t (m) for x t (m). When we differentiate this new family of Radon-Nikodym derivatives with respect to t, we find, pointwise for all t ≥ 0 and m ∈ M thatṙ A continuously differentiable function h : R + → R may have a set of zeros that is quite messy, but its positive part can still be expressed as an integral in a straightforward way. For applications of Theorem 2.10, the following elementary fact from real analysis is helpful.
Lemma 2.11. Suppose that the function h : R + → R is an absolutely continuous function with h(0) = 0 and derivativeḣ defined Lebesgue-a.e. Write J for the indicator function of the set {t ∈ R + : h(t) > 0}. Then, for all t ∈ R + , Proof. The fundamental theorem of calculus holds for absolutely continuous functions and so h(t) = t 0ḣ (s) ds for all t ∈ R + . Note that the function t → g(t) := h(t) + is also absolutely continuous. Therefore, the nonnegative function g is differentiable almost everywhere with respect to Lebesgue measure and g(t) = t 0ġ (s) ds for all t ∈ R + .
Choose a point t ∈ R + in the set of full Lebesgue measure where h is differentiable. Consider the following four alternatives.
(1) If h(t) > 0, then g(s) = h(s) for s in a neighborhood of t, and so g is differentiable at t andġ(t) =ḣ(t).
(2) If h(t) < 0, then g(s) = 0 in a neighborhood of t, and so g is differentiable at t andġ(t) = 0. (3) If h(t) = 0 and t is not an isolated point of the set {s ∈ R + : h(s) = 0}, then there exists a sequence (t n ) n∈N such that t = lim n→∞ t n with t n = t and h(t n ) = 0. Then, converges to zero for any sequence t j converging to t. For any j, g(t j ) either equals h(t j ) or equals zero, so g(t j )/(t j − t) also converges to zero, meaning that g is differentiable at t andġ(t) = 0. (4) If h(t) = 0 and t is an isolated point of the set {s ∈ R + : h(s) = 0}, then g need not differentiable at t, but the set of such t is countable and hence Lebesgue null.

Mutation measures with infinite total mass
The assumption that the mutation measure ν has a finite total mass underlies our development of the model, our use of Wasserstein metrics, and our main theorems in Chapters 5 through 8. However, the dynamic system itself, taken in isolation, can be defined for some cases when ν(M ) is infinite and the selective cost S is chosen suitably. For example, the case where ν is a multiple of Lebesgue measure on the positive reals is important for demographic applications. This section sketches how the dynamical system for such measures can be handled by viewing the state space of the dynamical system as a subset of L ∞ (M, ζ) for a suitable reference measure ζ.
• WriteH for the space of signed measures on M such that the trace on any bounded set Borel set A ⊂ M belongs to H. The spaceH is equipped with the σ-field generated by the maps π → π(A) as A ranges over the bounded Borel subsets of M. (The trace of a measure π on a Borel set A is the measure π(· ∩ A).) • Denote the subset of nonnegative measures inH byH + .
• Denote the set of integer-valued elements ofH + byḠ.
Fix ζ ∈H + , and identify L ∞ + (M, ζ) withK + , the set of elements ofH + that are absolutely continuous with respect to ζ with essentially bounded Radon-Nikodym derivatives.
If π ∈H + , then the Poisson random measure X π with intensity measure π takes values in the setḠ almost-surely. Suppose that the selective cost function Suppose that the selective cost is such that is a well-defined and Borel measurable map fromK + × M to R + .
Remark 2.13. An examination of the proofs in Section 2.5 of Theorems 2.9 and 2.10 shows that they apply in this broader setting and do not depend on the finiteness of ν(M).

Equilibria
One of the primary problems concerning our dynamical systems is to understand their asymptotic behavior over time. We begin the analysis of these asymptotics by identifying when fixed points of the dynamical system exist, and then examining whether there is convergence to these fixed points from suitable initial conditions.
We assume throughout this chapter that the assumptions of Theorem 2.4 always hold.
Definition 3.1. A finite fixed point or equilibrium for the dynamical system (we use the terms interchangeably) is a measure ρ * ∈ H + at which the driving vector field vanishes. That is, ρ * is absolutely continuous with respect to ν, with Radon-Nikodym derivative satisfying We introduce the terms box-stable and box-attractive when the above definitions hold if "neighborhoods" in the above definitions are replaced by sets of the form whereρ ≤ ρ * ≤ρ , and both measuresρ − ρ * andρ − ρ * are mutually absolutely continuous with respect to ν.
Remark 3.2. Note that box-stability (respectively, box-attractivity) is a weaker condition than stability (respectively, attractivity), because boxes do not contain open neighborhoods in the topology induced by the Wasserstein metric (that is, the topology of weak convergence), but open neighborhoods do contain boxes.
Of course, it is not obvious that the dynamical system needs to have fixed points, since ν could dominate all fitness costs. This possibility is easy to see in the one-dimensional case (when M is a single point), which we described in Section 1.2 and which we revisit in Section 3.1. However, we show in Section 3.4 that at least one fixed point exists when the mutation measure ν is small enough. In order to go further, we need to impose additional assumptions. For the case of multiplicative selection costs, Section 3.2 gives a complete description of the fixed points, of which there can be 0, 1 or 2.
In the remainder of this chapter we then impose a weaker assumption, namely that the selective cost is concave, in the sense that the marginal cost of an additional mutation decreases as more mutations are added to the genotype. (The formal definition is given as the last condition in Theorem 3.12.) Under this assumption, we show in Section 3.5 that trajectories starting from 0 increase monotonically, and we give a sufficient condition for them to converge to a finite equilibrium. This equilibrium is dominated by any other equilibrium. If the driving vector field at nearby points above this minimal equilibrium are in the negative "orthant", then trajectories starting above the equilibrium converge down to it, so the minimal equilibrium is box attractive. We give an iterative procedure for computing the minimal equilibrium that avoids following the dynamical system to large times in Section 3.7. Moreover, the iteration is a useful theoretical tool in Section 3.8, where we establish that the equilibria shown to exist for suitably small ν in Section 3.4 are in fact the same as the minimal equilibria in the concave setting and that these equilibria are then box stable. Finally, in Section 3.9 we apply the above results to the demographic selective cost introduced in Section 1.4 These results cover many cases of substantive interest, although they do not settle all relevant questions. Cases with very small ν(M) do not wholly put on display the rich array of differences between the full non-linear model and non-epistatic additive models. However, the conditions of Corollary 3.17 and Theorem 3.18 can often be verified in specific cases, and we have found that they can hold when ν(M) is only moderately small.

Introductory example: One-dimensional systems
Suppose as in Section 1.2 that selective costs depend only on the number of mutations and the original space of loci has been pushed forward into a space M consisting of a single point. The space of genotypes G may be identified with nonnegative integers N 0 , the selective cost S is simply an increasing function from N 0 to R + , and the mutation measure ν is a positive constant. The space H + of finite measures on G may also be identified with R + . The dynamical system (ρ t ) t≥0 is R + -valued and satisfies The system has equilibria at solutions to the equation ρF ρ = ν and we discussed some special cases in Section 1.2. In general, it is possible to construct selective costs for which the number of equilibria is arbitrarily large for a given mutation rate. For example, suppose the selective cost has magnitude 1 for 1 to 5 mutations and magnitude 2 for 6 or more. The resulting function −ρF ρ is shown in Figure 3.1. The number of equilibria may be 0, 1, 2, 3, or 4, depending on the value of ν. However, we can say quite generally the following things about one-dimensional systems.
• As long as S is not identically 0, the same is true for F ρ , so there is at least one equilibrium for ν sufficiently small. • The smallest equilibrium is attractive, unless it corresponds to a local minimum of −F ρ ρ, in which case it attracts only trajectories coming from below; trajectories starting above the equilibrium are repelled. • If S is bounded above, then −F ρ ρ is bounded below, and so there is no equilibrium for ν sufficiently large.

Introductory example: Multiplicative selective costs
It is probably apparent that one cannot hope to get anything like a closed-form solution for the dynamical system (2.10) for general selective costs. However, we show in this section that it is possible to solve (2.10) explicitly when the selective cost has the multiplicative form for some θ : M → R + . Moreover, in this case it is possible to analyze whether lim t→∞ ρ t exists, and we couple this analysis with a comparison argument in later sections to give sufficient conditions for the existence of a limit for more general selective costs. Suppose that ρ 0 is absolutely continuous with respect to ν. It follows from Theorem 2.10 that the measure ρ t has a Radon-Nikodym derivative m → r t (m) against ν for all t ≥ 0 such that for each m ∈ M the function t → r t (m) is continuously differentiable and solves the equation Substituting from (3.5), we see that the expected marginal cost is given by the product of 1 − e −θ(m) and E[exp{− m ∈g θ(m )}]. The first factor 1 − e −θ(m) is S(δ m ), the selective cost of a singleton mutation, which is the same as F 0 (m). The second factor, the expectation, can be evaluated via the Poisson identity (1.13) and it equals exp{− (1 − e −θ(m ) ) dρ t (m)}.
We now define analogues of moments for k ∈ N by the equations Our differential equation becomes This is an ordinary differential equation for each m ∈ M whose solution, a special case of (2.15), is given by Thus, we have reduced what is in general an infinite collection of coupled ordinary differential equations to the problem of finding two functions, b 0 and b 1 .
Of course, this benefits us only if we have an autonomous system of equations in just the functions b 0 and b 1 . Combining (3.7) with (3.6) we have We introduce three generating functions The system of ordinary differential equations (3.8) then becomes the first order, linear partial differential equation We may find the general solution of this PDE via the method of characteristic curves. Once we obtain the general solution (which involves the unknown functions b 0 and b 1 ), we have to impose the boundary conditions and to select the solution to our original differential equation.
Using Mathematica, the PDE is seen to have the solution Therefore, the functions b 0 and b 1 solve the following system of equations expressed in terms of the generating functions A(z) and C(z) and their derivatives A (z) and Note that this system of ordinary differential equations is autonomous. It depends only on the unknown functions b 0 and b 1 themselves, together with the known functions A (determined by the given mutation and selection) and C (determined by the initial conditions). Hence, these equations could at least be solved numerically.
(Of course, we started with a system with as many equations as there are points in M. This reduction to a system of two coupled ordinary differential equations is only advantageous if the space M has more than two points.) We can now obtain a necessary and sufficient condition for the existence of an equilibrium for (2.10) in H + when the selection cost is the multiplicative one of (3.5). Note that if ρ * is an equilibrium, then we have and so ρ * has a Radon-Nikodym derivative r * with respect to ν that satisfies ). Although the equation for r * sometimes makes sense even when ν and ρ have infinite total mass, as discussed in Section 2.6, here we are only considering ν and ρ in H + , the space of finite positive measures. An equilibrium exists in this space if and only if, first, and, second, the conditions defining b * 0 and b * 1 are consistent with the formula for r * , so that there is a constant c > 0 such that Note that there are three possible cases.
• If ν(M) < e −1 , then there are two equilibria, corresponding to the two distinct solutions of ce −c = ν(M). The equilibrium corresponding to the smaller c is attractive, while that corresponding to the larger c is unstable.
then there is a single equilibrium, which is unstable.
We are now in a position to give an example which previews the general approaches in the following sections. Take M = [1, ∞) ⊂ R + . We generate multiplicative selective costs from the function θ(m) = log(m/(m − 1)), so that We take mutation measure to be the measure ν(dm) = u(2/m 3 ) dm with total mass ν(M) = u. We have just seen that when u < e −1 there exists an equilibrium whose Radon-Nikodym derivative with respect to ν is given by Here c(u) is the smaller root of the equation ce −c = u.
The equilibrium Radon-Nikodym derivative is not bounded. It increases linearly with m ∈ M, even though the equilibrium measure ρ * (dm) = r * (m)ν(dm) has finite total mass and does belong to H + . If we want a bounded continuous function of m ∈ M, we need to turn to the product S(δ m )r * (m). In this example the product equals exp{c(u)}1, where 1 is the function taking the value 1 for every m ∈ M.
For every m ∈ M, the product is a continuously differentiable function of the total mass u of the mutation measure and is in fact for small u the solution of the differential equation If we did not know r * , we could arrive at this equation by differentiating (3.9). We exploit this approach to give a general construction of equilibria for small ν(M) in Section 3.4. The scale factor in this example with multiplicative costs can be obtained by iteration of the map c → ue c starting from c = 0. This scalar-valued iteration corresponds to an iterative scheme for generating r * itself, namely r → 1/F rν . For selective costs which are concave, in a sense defined in Section 3.5, an iterative scheme along these lines enables us to prove stability results for eqilibria in Section 3.8.

Fréchet derivatives
We need some machinery on derivatives of curves and vector fields in order to analyze equilibria in more generality. Consider a Banach space (X, · X ) and a closed convex cone X + ⊆ X. For x ∈ X + , let U x be the closed convex cone . Extending the usual definition slightly, we say that a map Φ : The definitions depend on the choice of cone. We shall be applying them to the cone H + and taking care to restrict directions to such positive measures.
As usual, we say that a curve ψ : For the sake of completeness, we record the following standard fundamental theorem of calculus and chain rule.
Lemma 3.3. Consider an interval I ⊆ R and a Banach space (X, · X ). Suppose that a curve ψ : I → X is differentiable at every t ∈ I and the curve t →ψ t is continuous. Then, Lemma 3.4. Consider an interval I ⊆ R, two Banach spaces (X, · X ) and (Y, · Y ), and a closed convex cone X + ⊆ X. Suppose for some t ∈ I that the curve ψ : I → X + is differentiable at t ∈ I and the map Φ : We also have a particular analogue of the product rule. Recall that H is the Banach space of finite signed measures on M equipped with the Wasserstein metric, and H + is the cone of positive measures within H.
• Write C b (M, R) for the Banach space of bounded continuous functions from M to R equipped with the supremum norm. • For η ∈ H + and m , m ∈ M, define the kernel Lemma 3.6. Consider an interval I ⊆ R and two curves γ : I → H + and f : Suppose that γ and f are differentiable at t ∈ I. Define a curve β : I → H + by β u := f u · γ u , u ∈ I; that is, β u is the element of H that has Radon-Nikodym derivative f u with respect to γ u . Then, β is differentiable at t witḣ Proof. This follows from Lemma 3.4 with and the curve ψ is differentiable at t withψ t = (ḟ t ,γ t ). Both proofs are straightforward and we leave them to the reader.
By our standing assumption, the conditions of Theorem 2.4 are in place. By the Lipschitz condition on S, with constant σ, via Lemma 2.5, the absolute value of the sum of the first two terms inK is bounded by σ δ m Was = σ and the absolute value of the sum of the second two terms is bounded by σ δ m Was = σ. No separate uniform bound on S is required. Thus, the map (η, m , m ) →K η (m , m ) is bounded by 2σ. Ideas similar to those behind Lemma A.12 and Lemma 2.6 establish that this kernel gives the Fréchet derivative of the map η → F η (·).
Moreover, straightforward coupling arguments establish the following bounds, where we recall that the constant σ is such that |S(g) − S(h)| ≤ σ g − h Was for all g, h ∈ G.

Existence of equilibria via perturbation
We now proceed to prove the existence of equilibria when the total mutation rate is sufficiently small. We fix a selective cost function S and define a family of mutation measures ν (u) for u ∈ R + and some ζ ∈ H + by In other words, the mutation measures are scalar multiples of each other differing only in total mass. A family of corresponding dynamical systems starting from the null state is given by An equilibrium for any one of these systems has a Radon-Nikodym derivative with respect to ζ.
Theorem 3.9. Consider a selective cost function S with S(0) = 0 that satisfies the monotonicity and Lipschitz conditions of Theorem 2.4. Suppose, moreover, that Then, there exists U > 0 and a curve (u, Proof. If measures ρ (u) do exist satisfying the conditions of the theorem, then we expect their Radon-Nikodym derivatives to satisfy an equation obtained by differentiating both sides of Taking advantage of our expression for the Fréchet derivative of F ρ and applying Lemmas 3.6 and 3.7 with η = p (u) ζ and η = dp (u) du ζ, the desired relationship comes out to be Here 1 ∈ C b (M, R + ) is the function with constant value 1.
Our strategy is to start with an operator equation of similar form, show that it does have solutions in C b (M, R + ), derive a differential equation from them, solve it, and identify p (u) with a suitable function of the solution. For Let D be the set of functions p ∈ C b (M, R) such that T p is invertible. It follows that 0 ∈ D, because T 0 (q)(m ) = S(δ m )q(m ) and inf{S(δ m ) : m ∈ M} > 0 by assumption. A standard result in operator theory (see Lemma VII.6.1 of [DS88]) tells us that the invertible operators form an open set in the operator norm topology, so that D includes all p such that T p − T 0 is sufficiently small, where (here only) · denotes the operator norm. By Lemmas 3.8 and 2.6 we can bound T p − T 0 in terms of the Lipschitz constant σ on S by We conclude that D includes an open ball around 0.
Define a map L : Thus, is the diagonal operator that multiplies a function of m ∈ M by the bounded function m → 1/S(δ m ) = L(0), and we can combine our bound on the operator norm of T p − T 0 to show that for a suitable constant c By requiring p ∞ to be small enough, we can make the first term on the right arbitrarily small and make 1−c p ∞ L(0) ∞ arbitrarily close to 1, so we can make Furthermore, again by Lemma VII.6.1 of [DS88], within this neighborhood, the map from an operator to its inverse is a homeomorphism and It follows that L(p) satisfies a Lipschitz condition. Then, by standard results on existence and uniqueness of solutions to ordinary differential equations in a Banach space, the ordinary differential equation dp (u) du = L(p (u) ) with initial condition p (0) = 0 has a solution on an interval [0, U ] and this solution takes values in C b (M, R + ). Thus, for 0 ≤ u ≤ U , p (u) = u 0 L(p (v) )dv satisfies the requirements in the conclusion of the theorem.

Concave selective costs
For an important class of examples, including the demographic example of Section 1.4, the selective costs are concave, in the sense that the marginal cost of adding a given mutation becomes smaller, the more other mutations are already present. Formally, this is stated in Definition 3.10. Under a few mild constraints, we show in Theorem 3.12 that concave selective costs yield monotonic solutions (ρ t ) t≥0 when started from the pure wild type population ρ 0 = 0, and hence such systems must either diverge to a measure with infinite total mass or converge to an element of H + . Corollary 3.17 gives a further condition that is sufficient to ensure that the limit is an element of H + . Our conditions for monotone increase in ρ over time turn out to be satisfied quite generally for the applications we have investigated. The conditions for the existence of a limit in H + , on the other hand, are not always satisfied, and there are important cases (discussed in [WES08]) for which ρ t increases to a measure with infinite total mass.
Lemma 3.11. A selective cost S is concave if and only if for all g ∈ G and m, m ∈ M. Moreover, if S is concave, then for π, η ∈ H + and m ∈ M.
Proof. Since elements of G have finite integer mass, we can prove (3.13) by induction on n := h(M) ∨ k(M). Our assumption is equivalent to (3.13) in the case n = 1. Assume now that (3.13) holds whenever h(M) ∨ k(M) ≤ n − 1. Suppose h(M) = n and k(M) ≤ n − 1. Let m ∈ M be in the support of h, and let h =h + δ m . Then Sinceh and k both have mass smaller than n, each of the terms in brackets is ≤ 0 by the induction hypothesis. To complete the induction, we need only address the case when h(M) = k(M) = n; this case proceeds exactly as above.
Finally, the Poisson random measures X π+η is distributed like the sum of independent copies of the Poisson random measures X π and X η , and so Theorem 3.12. Fix a mutation measure ν ∈ H + and a selective cost S : G → R + , that satisfies the conditions for η ∈ H + and m ∈ M. By Lemmas 2.5 and 2.6, thanks to the Lipschitz bound on S, it is clear The curve ρ is differentiable at each t ≥ 0 and satisfies The right-hand side is continuous in t. By Lemma 3.3, it then suffices to show thaṫ ρ t ≥ 0 for all t ≥ 0. By Lemma 3.4 and Lemma 3.7, the curve The value of this derivative at m equals the integral MK ρt (m , m )ρ t (dm ), where the kernelK is defined in Notation 3.5. By the concavity condition on S, for all m and m in M, Furthermore, by Lemma 3.6, Suppose now thatρ 0 ≥ 0. Define a negatively-rescaled version of t →ρ t by Then, By assumption, Writing out the derivative in terms of the kernelK and expressingρ t in terms of γ t , we obtain Let γ s = γ + s − γ − s be the Hahn-Jordan decomposition of γ s into its positive and negative parts. Since −K and the exponential factors are nonnegative, γ s can be replaced by its positive part in the inequality for γ t (B) thanks to the upper bound on −K. The inner integral over m is bounded above by γ + s (M)F ρs (m ). Hence, For any positive T , set C T equal to the product of two bounds, namely, first, the bound on the exponential multiplier and, second, the bound of the integral Both bounds hold for s ≤ t ≤ T by Lemma 2.5, and ρ s (M) ≤ ρ 0 (M) + sν(M) because F and ρ are nonnegative in (3.14).
Since C T is finite, we have shown that where the supremum is taken over Borel sets including the null set, so that β t is nonnegative. Now, γ + s (M) = β s . Hence, we have shown that It follows that the measure γ t is nonpositive. Thus,ρ t (which differs from γ t by a strictly negative Radon-Nikodym factor) is nonnegative. This finishes the proof of the claim for the caseρ 0 ≥ 0.
Ifρ 0 ≤ 0, then we define γ t to be + exp t 0 F ρs ds ρ t , and the rest of the proof carries through as before.
The following Comparison Lemma, also proved via Gronwall's Inequality, is a powerful tool for applications. It treats pairs of solutions ρ t and ρ t in which, informally speaking, ρ starts ahead of ρ and the marginal selective costs slowing the progress of ρ are always less than the marginal costs slowing ρ . In such a race ρ always keeps the lead.
Lemma 3.14. Consider two selective cost functions S and S that satisfy the conditions of Theorem 3.12. Let ρ and ρ be the corresponding solutions of (2.10).
We seek to prove that the positive part ξ + t is zero for all t ≥ 0.
where the supremum is over the Borel subsets of M. Here · T V is the total variation norm, which, as remarked in Section 2.1, dominates the Wasserstein metric on H. The function β is nonnegative, and we proceed to show that it is actually zero, so that there is no Borel set on which ρ is bigger than ρ .
Let (m, ρ) → F ρ (m) be the expected cost function corresponding to S , and (m, ρ) → F ρ (m) be the expected cost function corresponding to S . As before, via Lemmas 2.5 and 2.6, for fixed ρ, F ρ (·) and F ρ (·) can be regarded as elements of the space C b (M, R) of bounded continuous functions on M equipped with the supremum norm. Because η ≤ ρ , our assumptions imply the inequalities By Lemma 2.6 we have the Lipschitz bound By Lemma 2.5 we also have F ≤ 2σ. Bearing in mind that F is nonnegative, we see thatξ Thus, the inequality (3.17) onξ s implies Taking the supremum over Borel sets A on the left-hand side, we have a new constant K such that Gronwall's Inequality (Appendix A.1) then gives β t = 0 for all t in the interval [0, T ], and hence for all t ≥ 0. Thus, ρ t ≤ ρ t for all t ≥ 0.
An alternative proof along the same lines takes advantage of the L ∞ norm. It is crafted so that it does not depend on the assumption of finite total mass for ν.
Lemma 3.15. Consider two selective cost functions S and S and corresponding solutions ρ and ρ under the conditions of Theorem 2.9. Suppose for all g ∈ G and m ∈ M that Proof. Set η t = ρ t ∧ ρ t . Theorem 2.10 supplies functions (t, m) → r t (m) and (t, m) → r t (m) such that m → r t (m) and m → r t (m) are the Radon-Nikodym derivatives of ρ t and ρ t with respect to ζ = ρ 0 + ν for each t ≥ 0, and t → r t (m) and t → r t (m) are continuously differentiable for all m ∈ M. Write q ν for the Radon-Nikodym derivative of ν with respect to ζ. For π ∈ K + , let F π and F π be the expected cost functions corresponding respectively to S and S . Set , which, by Theorem 2.10 is Borel measurable. We now appeal to Lemma 2.11. Separately for every m ∈ M for all t ≥ 0 We write the integrand as the sum of three terms as followṡ The third term is never positive, since J(s, m) vanishes whenever r s (m)−r s (m) is negative. The second term is never positive, since the assumed inequality on the marginal costs makes F ρ s (m) ≤ F ρ s (m) for all m ∈ M and the concavity condition arranges for η s ≤ ρ s to imply F ηs (m) − F ρ s (m) ≥ 0 for all s ≥ 0 and m ∈ M. In contrast, the first term is never negative and the factor J(s, m) is redundant, by the same concavity argument applied to ρ s .
The Lipschitz condition on F bounds the first term by the quantity By assumption, the contribution of the starting state x 0 is negative, so we conclude for all m ∈ M that Gronwall's Inequality (Appendix A.1) then forces β t ≡ 0 for t ∈ [0, T ] and so for all t ≥ 0.
Remark 3.16. The proof of Lemma 3.15 does not depend on the finiteness of ν(M) and the lemma remains valid in the broader setting developed in Section 2.6.

Concave selective costs: Existence and stability of equilibria
If the conditions of Theorem 3.12 hold, trajectories starting from 0 either converge as time goes to infinity to an equilibrium state in H + or diverge to a measure with infinite total mass. We therefore wish to consider conditions that ensure the existence of an equilibrium with finite total mass. One approach is to compare the concave selective cost to a multiplicative selective cost. This produces the small benefit over the general existence result of Theorem 3.9 of providing an explicit value of ν(M) that is small enough to guarantee the existence of finite equilibria.
(a) Suppose that the selective cost S : G → R satisfies the conditions of Theorem 3.12 and also satisfies the bound for all m ∈ M for some constant ξ > 0 and function τ : Suppose also that ρ 0 is the null measure 0 and ν(M) ≤ e −1 ξ. Then, there Conversely, if there exist some constant ξ and function τ such that the reverse inequality holds and ν(M) > e −1 ξ, then lim t→∞ ρ t (M) = ∞.
Proof. Consider part (a). Let ρ be the solution of (2.10) with selective cost with cξ = exp{c}ν(M) (such a c exists by the assumption that ν(M) ≤ e −1 ξ). It follows from the results of Section 3.2 that ρ t = ρ * * for all t ≥ 0. Apply Lemma 3.14 with S = S and ρ 0 = 0 to conclude that ρ t ≤ ρ * * for all t ≥ 0. It follows from Theorem 3.12 that ρ t ↑ ρ * as t → ∞ for some ρ * ∈ H + with ρ * ≤ ρ * * . Now consider part (b). We define ρ as before, with selective cost S , but with initial condition ρ 0 = 0. Lemma 3.14 implies then that ρ t ≥ ρ t for all t ≥ 0. We know that ρ t is increasing in t, and there is no finite equilibrium. Suppose R := lim t→∞ ρ t (M) < ∞. Then, for any Borel set A, the quantity ρ t (A) is increasing in t and bounded by R, so it converges to a limit ρ * (A). It is easy to check that A → ρ * (A) is a measure in H + and that ρ t converges to ρ * in the Wasserstein metric (that is, in the topology of weak convergence). From (2.10) we know that for any Borel set A, and any s < t, where the operator D is the analogue of the operator D when the selective cost S is replaced by the selective cost S . We conclude that ν(A) − D ρ u (A) ≥ 0 for all u ≥ 0, and so Since D ρ is continuous in ρ by Lemma 2.5, it follows that the integrand on the right-hand side converges to ν(A) − D ρ * (A), which must then be 0. Since this is true for all Borel sets A, it would follow that ρ * would be an equilibrium for the dynamical system with selective cost S , contradicting the fact that no such equilibrium exists.
The monotone growth of systems with concave fitness cost functions allows us to derive a simple sufficient condition for stability of the "minimum equilibrium" ρ * , that is, the fixed point to which the dynamical system converges when started from 0. We know that all trajectories that start strictly below ρ * converge asymptotically to ρ * . Since the vector field vanishes at ρ * , this implies that the derivative of the vector field −D ρ * F [η] · ρ * − F ρ * · η is nonpositive for all positive directions η. The equilibrium is box stable if this nonpositivity extends to a neighborhood of ρ * , which can be guaranteed if the derivative is actually bounded away from 0, measured by its Radon-Nikodym derivative with respect to ν.
(a) Suppose that the selective cost function satisfies the conditions of Theorem 3.12, and the curve (ρ t ) t≥0 , started at ρ 0 = 0, converges to a finite fixed point ρ * . Suppose further that Then, the fixed point is box stable. (b) Moreover, if M is compact and the equation has no solution η ∈ H + that is absolutely continuous with respect to ν, then ρ * is box attractive.
Proof. Consider part (a). Define an intensity measure ψ s which is offset above the equilibrium ρ * by a constant multiple of the mutation measure ν. That is, ψ s := ρ * + sν. Let φ s = ν − F ψs · ψ s be the driving vector field evaluated at the point ψ s . We express the derivative of φ s with respect to s in terms of the quantity whose infimum is bounded below in the definition of C ρ * . This quantity would be denoted by T p [1] · ν in the notation of (3.12) with ρ * = p · ν, but p does not necessarily belong to C b (M, R + ).
The derivative of φ s with respect to s is given by Lemma 3.8 guarantees that where σ is the Lipschitz constant for the selective cost S. By the same lemma, By Lemma 2.6, |F ρ * +sν − F ρ * | ≤ 8σsν(M).
The remaining two terms are the terms bounded below by the assumption that C ρ * > 0. We have shown that the derivative of φ s with respect to s has a Radon-Nikodym derivative with respect to ν bounded below for all m ∈ M by For small enough positive s this lower bound is positive. For such s, starting from φ 0 = 0, Applying Theorem 3.12, we see that when ρ 0 = ψ s , the trajectory ρ t is monotonically decreasing. Thus, for all times t ≥ 0 we have ρ t ∈ B(ρ * , ψ s ), where B(ρ * , ψ s ) is the box defined in (3.2). Now consider part (b). Suppose that there is no η ∈ H + that is absolutely continuous with respect to ν and satisfies D ρ * F [η] · ρ * + F ρ * · η = 0. For all s sufficiently small, the trajectory starting from ψ s is monotonically decreasing, and converges to an equilibrium ρ (s) * . If ρ * were not box attractive, then these equilibria would be distinct from ρ * = ρ (0) * . Since the ρ (s) * are all between ρ * and ρ * + sν, we know that ρ (s) * − ρ * is absolutely continuous with respect to ν and belongs to H + . Consider the measures π (s) , defined by .
These are probability measures on M. Since M is compact, the space of probability measures on M is also compact (recall that the Wasserstein metric induces the topology of weak convergence), so there is an accumulation point. Let s 1 , s 2 , . . . be a sequence converging to u, with π (si) → η. Since the vector field vanishes at all of these points, the derivative in direction η vanishes as well, contradicting our assumption, and hence proving that ρ * is in fact box attractive.

Iterative computation of the minimal equilibrium
The measure π ∈ H + is an equilibrium for the dynamic equation (2.10) if it is true that F π · π = ν. Recall that F π · π is the measure that has Radon-Nikodym derivative F π with respect to π. The equation implies that π is absolutely continuous with respect to ν with a Radon-Nikodym derivative p that satisfies Under the conditions of Theorem 3.12, if (3.18) has a solution, then it has a minimal solution that arises as the Radon-Nikodym derivative with respect to ν of lim t→∞ ρ t when ρ is the solution of (2.10) with ρ 0 = 0.
If one is only interested in finding the minimal equilibrium numerically, then it would be desirable to be able to do so without having to solve (2.10). An obvious approach to that problem is to define a sequence of functions p n : M → R + inductively by p 0 = 0 and By Lemma 3.11, it is clear that if S is concave, then F p ν ≥ F p ν for p ≤ p . Because p 0 = 0 ≤ p 1 , it follows that p 0 ≤ p 1 ≤ p s ≤ . . .. Moreover, if p * * is any solution of (3.18) then p * * has to be a fixed point of the map p → 1/F pν , so p n ≤ p * * for all n. If there is a solution p * * such that M p * * (m) ν(dm) < ∞, then p n ↑ p * ≤ p * * as n → ∞ for some function p * : M → R + such that p * ν assigns finite mass to M. Thus, if a limit exists in H + for ρ t started from ρ 0 = 0 as time goes to infinity, it is greater than or equal to the iterative limit p * ν. But such a limit over time is also less than or equal to any other equilibrium by Lemma 3.14, the Comparison Lemma. So, starting from zero, when the iterative limit and the limit over time exist, they are the same limit, the so-called "minimal equilibrium".
Remark 3.19. If π ∈ H + is an equilibrium for (2.10) with Radon-Nikodym derivative p against ν, then, from (3.18), under the concavity assumption. Thus, a necessary condition for the existence of an equilibrium in H + is that the last integral is finite.

Stable equilibria in the concave setting via perturbation
Suppose that the selective cost is concave. In Section 3.4 we constructed equilibria for sufficiently small mutation measures. We know from Section 3.5 that all trajectories starting below an equilibrium ρ * * converge asymptotically to an equilibrium that is also dominated by ρ * * . This leaves open the question of whether the equilibrium constructed by perturbing the dynamical system away from ν ≡ 0 is the same as the minimal equilibrium ρ * to which the system converges when started from 0.  solving the equation with p (0) ≡ 0. The measure p (u) ν ∈ H + is the minimal equilibrium for the system with mutation measure uν for all u ∈ [0, U ]. Furthermore, the minimal equilibria so realized for u < U are box stable.
(b) Moreover, if M is compact and the equation has no solution η ∈ H + with η absolutely continuous with respect to ν, then ρ * is box attractive.
Proof. Following the method of Section 3.7, we may compute the minimal equilibrium as the limit of the iteration with p 0 ≡ 0. Let q n be the corresponding iterates for vν, where 0 ≤ u < v.
The concavity of S implies via Lemma 3.11 that p n ≤ q n for all n and By Lemma 2.6 the factor F pn − F qn in the second term on the right is no greater than 8σ q n − p n ∞ ν(M).
We now need upper bounds on the reciprocals of F p and F q . Thanks to the lower bound on S(δ m ), the conditions of Theorem 3.9 are satisfied. That theorem provides a curve of equilibria for u in [0, U ] for some U > 0. To avoid confusion with the minimal equilibria that we are constructing here, denote these equilibria from Theorem 3.9 as x (u) (m)ν(dm). By Lemma 3.14, x (u) ≤ x (U ) , and because x (u) is a fixed point of the map p → u/F pν and 0 ≤ x (u) , we have p n ≤ x (u) ≤ x (U ) for all n. Similarly q n ≤ x (v) ≤ x U ) for 0 < u ≤ v ≤ U . It follows from (2.13) that For fixed U , write a for the right-hand side of the inequality (3.19), which is constant over values of m ∈ M, and write b = 8σν(M)a 2 u. We now have By choosing U in [0, U ] small enough and requiring 0 < u ≤ v < U , we can make b less than 1. Iterating while bearing in mind that q 0 − p 0 = 0, we find q n+1 − p n+1 ∞ ≤ (a/ (1 − b)) |v − u| for all n Hence, with c = a/ (1 − b), the corresponding minimal equilibria, say p * and q * , satisfy We now drop the stars subscripting equilibria and write u → p (u) for the curve of Radon-Nikodym derivatives of minimal equilibria whose dependence on u we have been investigating. We have just verified the Lipschitz condition for all m ∈ M. The condition guarantees that u → p (u) (m) is Lebesgue almost everywhere differentiable, and is the integral of its derivative. Since p (u) satisfies the relation p (u) F p (u) = u for every u < U , we see by differentiating with respect to u that p (u) satisfies the differential equation in the statement of the theorem. By standard uniqueness results for ordinary differential equations, it is the unique solution with p (0) = 0. It therefore agrees with the solution provided by Theorem 3.9, here called x (u) but there (and henceforth) called p (u) , defined for all u in [0, U ].
It remains to verify box stability and box attractivity. Fix u ∈ [0, U ). The dynamical system with mutation measure U ν converges to a finite equilibrium p (U ) ν ∈ H + , where p (U ) ≥ p (u) . Let (ρ t ) t≥0 be the dynamical system with mutation measure U ν started from ρ 0 = p (u) · ν and let (ρ t ) t≥0 be the dynamical system with mutation measure uν started from ρ 0 = p (U ) · ν. Thus, (ρ t ) t≥0 starts below its equilibrium, and (ρ t ) t≥0 starts above its equilibrium. We havė Therefore, p (U ) > p (u) , and we can conclude from Theorem 3.12 that the measure ρ t stays bounded between the measures p (u) · ν and p (U ) · ν for all times t ≥ 0, and hence the minimal equilibrium p (u) · ν is box stable. Now consider part (b). We know that the system with mutation measure uν converges monotonically downward to an equilibrium when started from p (v) ν. The final part of the proof, showing that this equilibrium is in fact p (u) · ν, proceeds exactly as in the previous proof of Theorem 3.18.

Equilibria for demographic selective costs
As an example of Corollary 3.17, suppose that S is the demographic selective cost of Section 1.4, so that Here, as before, θ specifies an increment to the cumulative hazard function. This selective cost is concave, allowing us to capitalize on the results of Sections 3.5 and 3.8 so long as the θ are such that the required Lipschitz condition on the selective cost is satisifed, a requirement that is messy to formulate in general but easy to verify for most specific applications.
Lemma 3.21. Suppose that the selective cost function S is the demographic cost of (3.20). Suppose also the age-specific profiles θ satisfies the following two conditions. Proof. For all m ∈ M, We also have S(g + δ m ) − S(g) It follows from the first implication in Corollary 3.17 that the dynamical system started from the null state with these costs converges to a finite equilibrium ρ * in H + . In other words, if there is a range of fertile ages over which all deleterious mutations reduce survival by at least some minimal amount, then selection keeps the total intensity from going to infinity provided ν(M) ≤ e −1 ξ.
The converse implication in Corollary 3.17 cannot be readily applied to this setting, at least not with a constant function τ . However, in some cases we can give a direct proof that no equilibrium exists with finite total mass.
For example, suppose that S is the demographic selective cost of Section 1.4, with M = [α, β] for 0 < α < β < ∞, ν is a constant multiple of Lebesgue measure on M, f x is constant, and θ(m, x) = η1 [m,β] (x) for some constant η. Such a simplified model has featureless fertility between two ages that represent the onset and end of reproduction, mutations associated with effects at specific ages, constant mutation rate during the reproductive span, and equal increments to the hazard from all mutations. Similar models were introduced and studied by Brian Charlesworth in [Cha01]. For a suitable constant K , The bound on the right-hand side is independent of g, so the expected value of the marginal cost has the same bound. Any equilibrium ρ * must satisfy ρ * = ν/F ρ * . Since an equilibrium with finite total mass does not exist. Of course, (3.18) may have a solution p such that p·ν has infinite total mass and so does not belong to H + . In the setting of Theorem 3.12, the measure π = p·ν could still arise as the increasing limit of the solution ρ t to the dynamic equation (2.10) and be such that the population average selective cost E[S(X π )] would be finite. These more delicate possibilities are treated in [WES08]. We now investigate the box stability and box attractivity of the minimal equilibrium.
Theorem 3.22. Suppose the selective cost function S has the demographic form specified in (3.20), (or in (1.11) and (1.12)) with ρ 0 = 0 and f x dx < ∞. Suppose the age-specific effect profiles θ are such that the selective cost satisfies the Lipschitz condition of Theorem 2.4 and also satisfies the following two conditions.
• The infimum inf m∈M,x∈B θ(m, x) is strictly positive for some Borel set B ⊆ R + such that the integral B f x dx is strictly positive.
Proof. For a measure ρ ∈ H + , we write A ρ (x) for the aggregate population net maternity function, that is, The expected marginal cost F ρ is given by and the kernelK for computing its Fréchet derivative defined in Notation 3.5 turns out to beK The existence of a finite minimal equilibrium ρ * ∈ H + when ν(M) < ξ/e is guaranteed, as we have seen, by Corollary 3.17. For sufficiently small values of ν(M), ρ * is the same as the equilibrium guaranteed by Theorem 3.9 because we can bound S(δ m ) from below thanks to Lemma 3.21.
We derive an upper bound on ρ * . By hypothesis, ν(M) is strictly less than the smaller of ξ/e and 1/e. From the Comparison Lemma 3.14 and the discussion in Section 3.2 (substituting ν/ξ for ν), we know that ρ * is bounded above by an equilibrium solution with multiplicative costs. Specifically, where c is the unique solution in [0, 1] of the equation ce −c = z for z = ν(M)/ξ. As we have seen in Section 1.2, this solution exists when ν(M) ≤ ξ/e and can be expressed in terms of Lambert's W-function, dating back to 1758. Moreover, the map z → c is convex, lying below the line z → ez, so our hypothesis ν(M) < 1/e yields cξ < 1, implying e c ν(M) ≤ 1 − for some > 0. Since ρ * has a bounded Radon-Nikodym derivative, call it r * , with respect to ν, the function F ρ * (m) is bounded below, away from zero.
To establish the claims about stability and attractivity, we appeal to Theorem 3.18. Consider any measure η in H + which is absolutely continuous with respect to ν with Radon-Nikodym derivative q η (m), supplying a direction for a perturbation of ρ * . As before, we put ψ s = ρ * + sη and φ s = ν − F ψs · ψ s . The derivative at s = 0 of −φ s with respect to s is the measure T r * (q η )·ν in the notation of (3.12). Expressing the Fréchet derviative of F ρ * in terms of the kernelK, this measure is given by Examine the nonnegative function of m and x multiplying −ρ * (dm ) in the first term on the right-hand side. Because 1 − exp{−θ(m , x)} ≤ 1 − exp{−τ } for all m and x, this multiplier is bounded above by (1 − exp{−τ })η(M)F ρ * (m ). The nonnegative function multiplying η(dm ) is F ρ * (m ). Hence, Taking advantage of the bound on ρ * , in terms of Radon-Nikodym derivatives, We now use Theorem 3.18. For box stability we put η = ν, so that q η ≡ 1, and, as required, inf m∈M T r * (1)(m) is bounded below by inf m∈M F ρ * (m) > 0.
For box attractivity, when M is compact, we let η be any probability measure in H * . As required, the left-hand side of (3.22) cannot be almost-surely zero, since the integral over M of the right-hand side with respect to ν is bounded below by inf m∈M F ρ * (m) .
With demographic costs, assuming M to be compact is not an onerous restriction. Taking advantage of the boundedness of demographic costs, M can often be embedded in a compact space without disrupting the Lipschitz condition on S, and Theorem 3.22 then comes into force.

CHAPTER 4
Mutation-selection-recombination in discrete time We devote the remainder of this work to establishing rigorously the claim that we argued heuristically in the Introduction: The continuous-time dynamical system (1.8), which we defined formally in (2.10), is the limit of a sequence of discretetime infinite population mutation-selection-recombination models in the standard asymptotic regime where selection and mutation are weak relative to recombination and both scale at the same infinitesimal rate in the limit.
More specifically, we show that our continuous time model is a limit of the sort of discrete generation, infinite population mutation-selection-recombination models considered by Barton and Turelli [BT91] and Kirkpatrick, Johnson, and Barton [KJB02], once such models have been extended to incorporate our more general definition of genotypes. Such a result not only justifies our model as a tractable approximation to more familiar models in the literature, but, as we remarked in the Introduction, it also marks out the range of relative strengths for mutation, selection, and recombination where this approximation can be expected to be satisfactory, and thus where the resulting conclusions that underly applications to the demographic study of longevity can be trusted.

Mutation and selection in discrete-time
This section describes the operations of mutation and selection in the discretetime setting, leaving the description of recombination to Section 4.2.
We consider a sequence of models indexed by the positive integers. In the n th model, time is sped up by a factor of n, so that n generations pass in 1 unit of time. Unlike familiar diffusion approximations for genetic processes, all models in the sequence are infinite-population models; n does not have any reference to population size. The distribution of genotypes in the population is described, as before, by a probability measure on G, the space of finite integer-valued measures on the set M of loci.
At each generation each new birth gets a random set of extra mutations away from the ancestral wild type added to those mutant alleles it inherited from its parents. In the n th model in our sequence, the loci at which these mutations occur are distributed as the Poisson random measure X ν/n with intensity ν/n. The fitness of a genotype g ∈ G is e −S(g)/n . The n's in the denominators are what, in effect, speed up time. Mutation and selective cost per unit time are maintained. In a finite, non-zero amount of time the population is asymptotically subject to finite, non-zero "amounts" of mutation and selection.
We now define the four principal operators that act on the space of probabilities on G. The operators M n , S n , R, and P describe the transformation in genotype distribution by, respectively, one round of mutation, one round of selection, one round of recombination, and a process of complete Poissonization in which a genotype with possibly linked loci is replaced by one in which there is no linkage. The M n , S n , and P operators are defined in this section. The R operator, requiring further structure, is defined in the following section. The intuition behind these definitions is described in the Introduction. In this chapter we use the notation F without a subscript to stand for a bounded Borel test function. There is no connection with the subscripted function F π defined in Definition 2.3. We also use the letter Q to stand for a generic probability measure, so that P remains available for Poisson probability measures like the measures that describe the continuous-time dynamical system. Notation 4.1. Given a probability measure Q on G, define new probability measures M n Q, S n Q, and PQ by their action on a bounded Borel test function F : G → R + as follows We now show that as n becomes large, our discrete-time operators, considered separately, come to agree with their continuous-time counterparts. When n is large, P{X ν/n = 0} = exp{−ν(M)/n} ≈ 1 − ν(M)/n, and conditional on the event {X ν/n = 0} the random measure X ν/n is approximately a unit point mass with location in M chosen according to the probability measure ν(·)/ν(M). In other words, X ν/n takes the value δ m with probability approximately ν(dm) and takes the value zero otherwise. For any probability measure Q on G When we start with a population genotype distribution Q and assume that selection precedes mutation, the population genotype distribution after one generation of mutation and selection is M n S n Q. A trajectory of the resulting discretetime model is defined by iteration. That is, given an initial population genotype distribution Q 0 , the population genotype distribution after k generations is M n S n k Q 0 .
If, after speeding up time in the n th model, the resulting sequence of trajectories has a continuously differentiable limit P t , this limit should satisfy the equation for test functions F : G → R + of the form F (g) = exp{−g[f ]} for some Borel function f : M → R + . For this special choice of test function, this is precisely the equation defining dynamical system without recombination that we introduced in [SEW05] and derived heuristically in (1.5). In fact, these test functions are enough to consider, since they determine probability measures on G. Formally, a proof that the discrete-time dynamical system converges to this continuous-time one in the absence of recombination would require that we prove the existence of the continuously differentiable limit, a fact that we assumed above. We forgo such a proof, since we wish to incorporate recombination.

Recombination in discrete-time
We now introduce recombination. Imitating [BT91], we think of a recombination event as taking two genotypes g (1) , g (2) ∈ G from the population and replacing the genotype g (1) in the population by the genotype g defined by g(A) := g (1) (A ∩ R) + g (2) (A ∩ R ), where R ⊆ M is the particular segregating set for the recombination event. That is, the new individual with genotype g has the same accumulated mutations as the individual with genotype g (1) (respectively, g (2) ) for "loci" in the set R (respectively, R ).
As described in the Introduction, we are working in an abstract framework in which loci are just places at which mutations from wild type can occur rather than concrete physical loci per se. From a biological point of view, the recombination process might reasonably be defined on a linear sequence of loci, as in [BT91], with ends of chromosomes at pre-determined positions, and with recombination caused by crossovers during meiosis. Our abstract space of loci accommodates such a concrete representation but also leaves us freedom to specify loci using structures related to their demographic effects rather than their chromosomal positions.
Of course, we think of g (1) and g (2) as being chosen independently at random according to the particular probability measure describing the distribution of genotypes in the population. We also imagine that the segregating set is chosen at random via some suitable mechanism. In order to discuss random sets rigorously, we follow formalism described in [Ken74] and define a σ-algebra on sets of Borel subsets of M by the requirement that all incidence functions with Borel subsets are measurable. A consequence of this definition is that if A is a random Borel set and κ is a finite measure, then κ(A) is a real-valued random variable. We suppose that there is a probability measure R that describes the distribution of the random set of loci that segregate together. We always assume, without loss of generality, that R is symmetric in the sense that where R denotes the complement of the set R.
where Q is a Borel probability measure on G, the function F : G → R is bounded Borel, and B(M) is the collection of Borel subsets of M.
Thus, if Q describes the distribution of genotypes in the population, then RQ describes the distribution of a genotype that is obtained by picking two genotypes g (1) and g (2) independently according to Q, picking a segregating set R according to the probability measure R, and forming a composite genotype that agrees with g (1) on the set R and g (2) on the set R . Where confusion with subscripts might result, the customary notation g| R is also used for the restriction g R of g to the set R.
Recall from Notation 2.1 that µQ denotes the intensity measure of Q; that is, Remark 4.3. Note that µRQ = µQ; that is, the intensity measure of the random measure describing the genotype of a randomly chosen individual is left unchanged by recombination. Specifically, for Borel sets B, Moreover, if P is the distribution of a Poisson random measure, then RP = P .
We are now in a position to put the operators for recombination, mutation, and selection together to describe the whole discrete-time dynamical system.

Notation 4.4.
• Q 0 is the distribution of any simple, G-valued random measure and describes the distribution of genotypes in the initial population; • P 0 = PQ 0 = Π µQ0 = Π ρ0 is the Poisson probability measure with the same intensity as Q 0 ; • Q k := (RM n S n ) k Q 0 and describes the distribution of genotypes in the population after k generations of selection, mutation, and recombination; • P k := (M n ) k P 0 is the Poisson probability measure on G that results from k generations of mutation in the absence of selection starting from P 0 ; • O k := (PM n S n ) k P 0 is the Poisson probability measure resulting from k iterations of mutation and selection with intensity rescaled by 1/n and complete Poissonization (rather than recombination) after every round.

Recombination trees and annealed recombination
In order to establish the convergence of our discrete-generation dynamical system to our continuous-time dynamical system, we need to understand the effects of mutation, selection and recombination not only over a single generation, as we have done in Sections 4.1 and 4.2, but also over a number of successive generations. Over k generations, each of the operators M n , S n , and R acts k times. However, these operators are interleaved: there is first an S n , then an M n , then an R, then another S n , then another M n , and so on, and the operators in this product do not all commute with each other. For our convergence proof, we want to be able to decompose the impact of each kind of operator over k generations into its impact on the mutant alleles that first appear in a particular generation. Separating out the k-fold effects of each operator requires further structure to keep track of the generation in which a given mutant allele first appears. In this section we define structure for k-fold iterates of the recombination operator in terms of a version of a genealogical tree. In the next sections we define structure for separating out effects of M n and of S n over k generations by introducing a concept of "vintage", tracking the vintages of mutant alleles, and defining associated "starred" versions of our basic operators that are defined on genotypes that have been decomposed according to the allele vintages.
The iterates of our recombination operator implicitly create a genealogical tree. The genotype of an individual sampled from the population at generation k is formed from the genotypes of two parents sampled from the population, which are formed from the genotypes of four grandparents, and so on back to 2 k individuals sampled from the initial population at generation zero. Taken together, these 2 k+1 − 1 individuals comprise the nodes of a complete rooted binary tree. The root stands for the individual sampled at generation k and the leaves stand for the earliest ancestors back in the initial population at generation 0. Each node except the leaves has two edges (branches) directed in from parent nodes and each node except the root has one branch directed out to the offspring node.
For each choice of the time-scaling factor n, we have a finite number of generations k max = nT up to time T . The tree of ancestors for an individual in generation k just introduced may be envisioned as a subtree of the tree built from a root node at generation k max . Call this latter tree L.
The tree structure defines a partial ordering among nodes in the usual fashion. We label nodes in any convenient way, and write υ ≺ if node υ is an ancestor of node , that is, if there exists a path from υ to the root without duplicate edges containing . Nodes prior in the ordering are prior in time. We count generations starting with the leaves at generation zero. Since the letter "g" is already in use for genotypes, we write V ( ) for the generation ("vintage") of node .
The recombination structure is implemented by assigning complementary segregating sets B and B to the two branches connecting a node to its parents: the genotype of the corresponding individual in the set B (respectively, B ) coincides with that of the first (respectively, second) parent in the set B (respectively, B ).  The first picture shows in black the sets W (v, ) at the node v, for v in the first two ancestor generations preceding , so vintages k, k−1, and k−2. The second picture shows in gray the recombination set R (vintage k) and R and R (vintage k − 1) that generated these partitions. Note that W ( , ) = M. Note too that some sets in the partition will be empty, meaning that all genetic information from that ancestor has been lost through recombination.
More specifically, we first assign each node an independent set R with distribution R. (Sets for the leaves go unused.) Suppose that the parents of node are and . We label the branch connecting (respectively, ) to with the label (respectively, ). The complementary segregating sets assigned to the branches labeled and are, respectively, B = R and B = R . There is some ambiguity in this construction due to the choice of which branch receives R or its complement, and this can be resolved by embedding the complete rooted binary tree in the plane so that we can distinguish between the "left" and "right" ancestors of a node. Probabilistically, however, this ambiguity is immaterial thanks to the symmetry of the recombination measure -any way of resolving it will lead to an assignment of segregating sets to branches that has the same distribution. When and are ancestors (not necessarily parents) of a node and belong to the same generation, the sets W ( , ) and W ( , ) are disjoint subsets of M, since the branches leading to the first common descendant pass through sets R and R that are complements of each other. The sets W (υ, ) for fixed over all υ satisfying υ ≺ and V (υ) = j for some generation j cover M and so form a partition of M (We adopt the convention that a partition may contain empty sets.)  Here the mutation space M is the unit square, and the partition A is the collection of four sets {A 1 , A 2 , A 3 , A 4 } shown in the lower left. The mutations of four grandparent genotypes are shown at the top, and these genotypes (shown with a different symbol for each grandparent to clarify the origins of each one) are filtered through the partition to produce the offspring genotype at lower right.
We are now in a position to specify the combined action of k applications of the recombination operator.
Notation 4.6. For any finite partition A = {A 1 , . . . , A L } of M into Borel sets, some possibly empty, the annealed recombination operator R A is the operator acting on probability measures Q in G defined by the condition for bounded Borel functions F : G → R. Here, g| A is the restriction of the integervalued measure g to the subset A, also written g A as in Notation 4.2 in contexts where no confusion with indexing subscripts would result. See Figure 4.2.
In other words, if Y 1 , . . . , Y L are i.i.d. random measures with common distribution Q, then R A Q is the distribution of the random measure Y 1 | A1 + · · · + Y L | A L .
Via annealing, the genetic legacies from ancestors at generation zero generate the k-th iterate of the recombination operator as follows: Lemma 4.7. Let ∈ L be a node with V ( ) = k. Let A = (A 1 . . . A 2 k ) be the random partition of M consisting of the 2 k sets {W (α, ) : α ≺ and V (α) = 0}, constructed from the random segregating sets {R υ : υ }. Then, for any probability measure Q on G and bounded Borel function F : G → R, Proof. The claim holds trivially for k = 0. We proceed by induction. Suppose the claim holds up to generation k − 1 and thus for the parent nodes and of node . For all ω ≺ , either ω or ω , and W (ω, ) = W (ω, ) ∩ R or W (ω, ) = W (ω, )∩R , respectively. For the genotypes of the parent nodes, write Then, by the induction hypothesis, where each Q υ is a copy of Q, and the analogous equation holds for Γ and . Applying the operator R to the expressions for R k−1 gives Substituting for Γ and Γ shows this last expression is equal to the claimed formula.
We use the tree structure of L to construct a nested sequence of random partitions A 0 ( ) ⊇ A 1 ( ) ⊇ . . . ⊇ A V ( ) ( ) of M for an arbitrary node ∈ L. There is one set in the partition A j ( ) for each ancestor of the individual in generation j. This set represents the portion of that ancestor's genotype inherited by the individual through the shuffling together of genotypes due to recombination. Remark 4.9. Note that A k ( ) is a refinement of A k+1 ( ), and A V ( ) ( ) is the trivial partition {M}. Note also that for j ≥ k the distribution of the sequence of partitions (A 0 ( ), A 1 ( ), . . . , A k ( )) is the same for all nodes with V ( ) = j.
Recall that R is the recombination set at node . Suppose that has parents and and that the sets R and R are associated with the branches labeled and . Then, for k ≤ V ( ) − 1, Notation 4.10. Fix a node ∈ L with V ( ) = k. Write (A k,0 , A k,1 , . . . , A k,k ) for the nested sequence of random partitions (A 0 ( ), A 1 ( ), . . . , A k ( )).
Remark 4.11. The distribution of (A k,0 , A k,1 , . . . , A k,k ) does not depend on the choice of the node with vintage k. Note also that for 0 ≤ j ≤ k the sequence (A k−j,0 , A k−j,1 , . . . , A k−j,k−j ) has the same distribution as the sequence (A k,j , A k,j+1 , . . . , A k,k ). We can re-state the conclusion of Lemma 4.7 as

Vintages
The annealed recombination operator allows us to condition on choices of segregating sets while we isolate k-generational effects of mutation and selection. We next consider how the process of mutation relates to the recombination tree. Any element of M assigned mass by the genotype for a node at generation k has to have entered at some earlier generation, either in one of the nodes for ancestors in the initial population (distributed according to Q 0 ) or as a new mutation at one of the intermediate nodes in the tree of ancestors (distributed as a Poisson random measure with intensity measure ν/n). In our infinite-population setting, selection reweights the probabilities of outcomes. It does not remove elements of M or alter their pattern of descent from nodes of the tree. Although Q 0 may not be a Poisson probability measure, one of our assumptions will require it to be absolutely continuous with respect to the Poisson probability measure P 0 with the same intensity measure. It turns out that the difference between Q 0 and P 0 can be treated as a process of reweighting, analogous to the reweighting resulting from selection. With this strategy in mind, we use Poisson random measures to build a joint distribution of random genotypes across the tree, a template to which reweighting is applied.
Specifically, we assign to each node υ in L an independent Poisson random measure Γ υ with distribution P 0 = Π ρ0 when V (υ) = 0 and with distribution Π ν/n otherwise. Given a choice of the segregating sets R , which define the genetic legacies W at each node, consider restrictions of the Poisson measures to the legacy sets and put (4.12) Ξ = υ Γ υ | W (υ, ) .
These random elements of G are independent across nodes within any one generation but are dependent across nodes down any line of descent. They correspond to the genotypes of the ancestors of a randomly chosen individual in generation k max in the absence of selection starting from a population with genotype distribution P 0 . With this representation, which underlies the crucial Lemma 7.5 at the start of Chapter 7, we may separate out the effects of mutation over k generations from the effects of the other operators. Any mutant allele found in the genotype of a reference individual sampled from the population at generation k originates in the Γ υ term for some ancestral node. However, knowing the genotype at generation k and knowing the segregating sets for the tree of ancestral nodes is not enough to identify the origin of each mutation. A mutant allele found in the subset of M inherited from a particular great-grandparent might have originated from that great-grandparent or it might come from a new mutation in the same portion of the parental genome. What is missing is information on the generation, or vintage, from which the mutation originates. Because we need to be able to condition on the identities of the mutations that are still extant in the genome at generation k, while we average over all the other contributions to the Γ υ , we introduce additional structure to keep track of vintages.
We can rewrite (4.12) as The term separates out the contribution to Ξ due to mutant alleles that first appeared in generation j in the lineage of the individual associated with node . This suggests that we imagine that the genotype g of an individual in generation k is decomposed as a sum g = g 0 +· · ·+g k , where g j , j < k−1, is the contribution to g that first appeared in an ancestor in the past generation j and g k is the contribution of new mutations that first appeared in the individual herself. We thus have a richer structure where a genotype g ∈ G is replaced by the (k + 1)-tuple g = (g 0 , g 1 , . . . , g k−1 , g k ) in the product space G k+1 that decomposes g according to vintage. In most contexts there is no danger of confusing the subscripts for vintage components of g with subscripts for Borel sets to which genotypes g may be restricted.
The k + 1-tuple giving the decomposition according to vintage of an individual drawn at random from the population has a joint distribution which is a probability measure on G k+1 . This object describes the vintages of the current stock of mutations retained in the genomes of individuals in generation k. The force of selection does not depend on the decomposition by vintages -it only matters what mutations are present in an individual's genotype, not when they entered her lineage. But because selection alters the weights of whole genotypes in the population it necessarily alters the overall distribution of the decomposition by vintages. Recombination shuffles contributions within vintages. Mutation adds a new component to the decomposition by vintages at each generation.
In order to have a single space on which all our vintage-specific distributions are defined, we form the disjoint union G * of the spaces of tuples for successive generations starting with the initial population for k = 0 on up. That is, (4.14) where denotes disjoint union. In principle, a probability measure on this space can assign positive probability to tuples in several different generations and can thus be thought of as a sum of sub-probability measures on the different disjoint pieces. In practice, the probability measures that we consider are always concentrated on the tuples for some particular generation. We can always return from our big tuple space G * to G itself by adding up the components associated with each vintage. The operators for mutation, selection, and recombination M n , S n , and R, which we have defined to act on probability measures on G, all have counterparts acting on the probability measures on our big tuple space G * . Just as M n , S n , and R combine to take the distribution of genotypes in one generation to the distribution in the next, these new operators combine to take the distribution of genotypes decomposed by vintage in each generation into the distribution in the next. We denote these new operators by adding a star to the notation for the corresponding operators acting on probability measures on G.
In order to define these new operators, we first need to introduce a little more notation. We usually represent a generic element of G * by the boldface g. There are natural "projection" operations Σ : G * → G defined for g = (g 0 , g 1 , . . . g k ) ∈ G k+1 by Σ(g) := g 0 + g 1 + · · · + g k and In essence, Σ removes the labels that record the vintages of the various mutations from the ancestral wild type, while Ψ j isolates mutations with vintage j. Note that, as operators, Σ = ∞ j=0 Ψ j . Given a probability measure Q on G * , we define, with a slight abuse of notation, probability measures ΣQ and Ψ j Q on G by for a Borel function F : G → R + .
Although we shall only be employing probability measures on G * that are in fact concentrated on some G k , we give a brief account of the structure of a general probability measure on the disjoint union G * . As we remarked above, a probability measure Q on G * may be thought of as a sequence of sub-probability measures (Q (k) ) ∞ k=0 , where Q (k) is the portion of Q concentrated on (k + 1)-tuples from G. Thus, ∞ i=0 Q (k) (G k+1 ) = 1. We interpret such a Q as the distribution of a finite, random length sequence of random measures (X 0 , X 1 , . . . , X I ), with Q (k) (·; ·)/Q (k) (G k+1 ) being the joint distribution of (X 0 , X 1 , . . . , X k ) conditional on the event {I = k}, and Q (k) (G k+1 ) being the probability of the event {I = k}.
Notation 4.12. Consider a probability measure Q on G * that is concentrated on G k+1 for some k ∈ N. Write (X 0 . . . X k ) for a k + 1-tuple of random measures with conditional distribution Q (k) .
• Define a new probability measure M * n Q on G * to be the distribution of (X 0 . . . X k , Y ), where Y is an independent Poisson random measure with intensity measure ν/n. • Define a new probability measure S * n Q on G * that is concentrated on G k+1 by E[exp{−S(X 0 + · · · + X k )/n}] for any bounded Borel function F : G * → R + .
• Define a new probability measure R * Q on G * that is concentrated on G k+1 by F (g) dR * Q(g) := F (g (1) |A + g (2) |A ) dQ(g (1) ) dQ(g (2) ) dR(A) for any bounded Borel function F : G * → R + . • Define a new probability measure P * Q on G * that is concentrated on G k+1 by letting P * Q (k) be the distribution of (Y 0 , . . . , Y k ), where the random measure Y j is Poisson with the same intensity measure as X j and Y 0 , . . . , Y k are independent.
Observe that we have the four intertwining relations ΣM * n = M n Σ, ΣS * n = S n Σ, ΣR * = RΣ ΣP * = PΣ. These equalities confirm that starred operators agree with the unstarred ones once the labeling by vintage is removed.
Our initial conditions P 0 and Q 0 can also be regarded as probability measures on G * that are concentrated on sequences of length 1.
Notation 4.13. Set (4.15) The operator O * k is the analogue of Q * k with the complete Poissonization operator P * replacing the recombination operator R * .
Note that all three probability measures O * k , Q * k and P * k are concentrated on genotype sequences of length (k + 1). Note also that if (X 0 , . . . , X k ) is distributed according to P * k , then X 1 , . . . , X k are independent Poisson random measures, each with intensity measure ν/n, and X 0 is independent with distribution P 0 . As we expect, Q k = ΣQ * k , O k = ΣO * k , and P k = ΣP * k , where we recall that Q k = (RM n S n ) k Q 0 , O k = (PM n S n ) k Q 0 , and P k := (M n ) k P 0 .

CHAPTER 5
Shattering and the formulation of the convergence result

Shattering of random measures
We expect recombination to break up dependencies between different parts of the genome, so that when k is large R k P should be approximately Π µP , the distribution of the Poisson random measure with intensity µP in the notation established in Notation 2.1.
In order that this approximation should hold for a given probability measure P , it must generically be the case that there is a positive probability that the segregating set and its complement both intersect any set with positive µP mass in two sets that each have positive µP mass. The following condition (with λ = µP ) is key to establishing establishing quantitative bounds on the rate with which R k P converges to Π µP .
Definition 5.1. Given a (symmetric) recombination measure R and a finite measure λ on M, we say that the pair (R, λ) is shattering if there is a positive constant α such that for any Borel set A, We show in Section 5.3 that if the pair (R, µQ) is shattering and P satisfies a further simple condition, then R k Q converges to Π µQ as k → ∞.
The concept of shattering is illustrated by the following elementary example.
Remark 5.2. Suppose that N is a finite simple random measure on M = (0, 1]. We think of M in this case as a physical chromosome and N as the set of crossover points formed during meiosis. Write 0 < T 1 < . . . < T L < 1 for the successive points of M. Set T 0 = 0 and T L+1 = 1. Let Z be a {0, 1}-valued random variable that is independent of N with P{Z = 0} = 1 2 . Define R to be the distribution of the random set given by Take λ to be any diffuse probability measure (that is, λ has no atoms). Suppose that there is a constant c such that P{N (u, w] = 1} ≥ cλ((u, w]) for 0 < u < w ≤ 1. This is the case, for example, if N is Poisson random measure with intensity measure bounded below by a positive multiple of λ or N consists of a single point with distribution bounded below by a positive multiple of λ. Also, for most reasonable simple random measures with intensity Cλ for some constant C, it is the case that P{N (u, w] = 1} ≈ E[N (u, w]] = Cλ((u, w]) when |u − w| is small, where the notation ≈ indicates that the ratio of the two sides is close to 1.
For the recombination measure R in this example we have for the right-hand side of (5.1) As for the left-hand side of (5.1) observe that Thus, the pair (R, λ) is shattering with constant α = 3/c. for all r ∈ R + . Loosely speaking, this condition says that the probability that two loci inherit their contents from different parents dominates a multiple of the λ mass of a ball with radius the distance between the two loci. By a change of variables, Thus, (R, λ) is shattering with constant α = c −1 .
We collect some elementary observations about the shattering condition in the following lemma.
(e) For any Borel set A, Remark 5.5. Most pairs (R, λ) that arise naturally in applications are shattering, but there do exist pairs for which the recombination measure R splits every set with positive λ-measure into sets with positive λ-measure and the pair nonetheless fails to be shattering. For example, let M be the positive real axis. Let λ have a density with respect to Lebesgue measure equal to 2 −k on [k, k + 1). Consider random partitions of the axis into two parts [0, x) and [x, ∞) (taken for the sake of symmetry in random order), where the recombination measure chooses x according to a probability density equal to 2 −k (1/k)(x − k) −1+1/k on [k, k + 1). This density assigns the same measure to each whole interval [k, k + 1) as λ, namely 2 −k . For each k, consider the set A k = [k, k + 1). Put y = x − k. If the pair was shattering, there would be some constant α such that for all k With probability 1 − λ(A k ), the partition does not cut the interval A k , that is, x falls outside the interval, and the integrand on the right-hand side vanishes. The rescaled measure R(dR)/λ(A k ) restricted to A k is a probability measure, and (5.2) can be rewritten It follows that there is no finite shattering constant α which satisfies the required condition for all the sets A k , establishing that this pair (R, λ) is not shattering.

Consequences of shattering
We are now in a position to state and prove a pair of results that quantify the extent to which R k Q for some probability measure Q on G is a mosaic of many small pieces, each taken from genomes sampled independently from the population described by Q, where we measure the size of a subset of M using its µQ mass.
The following is a convenient way of measuring the extent to which a partition of M is made up of sets that each have small mass with respect to some reference measure. We apply it to the random nested partitions defined in Notation 4.8.
Notation 5.6. Suppose that A = {A 1 , . . . , A L } is a partition of M, λ is a measure on M, and r > 0. Set Lemma 5.7. Suppose that the pair (R, λ) is shattering with constant α. Then, for all k ∈ N 0 , Proof. Recall that A k+1,0 is the random partition for some ∈ L with V ( ) = k + 1 (the choice of is arbitrary insofar as the distribution of A k+1,0 is concerned). By construction, a set of the form W (υ, ) with V (υ) = 0 and υ ≺ can be written as either W (γ, ) ∩ R γ or W (γ, ) ∩ R γ , where V (γ) = 1 and γ ≺ , and so for j ∈ N 0 and recall the notation From the above and the symmetry of R, For any partition A = {A 1 , . . . , A L } of M, it follows from the Cauchy-Schwarz inequality that Thus, 2 . By construction, the random partition A 1 ( ) has the same distribution as A k,0 . Applying Jensen's inequality to the concave function x(1 − x/c), we see that We have X 0 = λ(M) 2 , which is strictly smaller than c = αλ(M) since the shattering constant α is necessarily greater than λ(M). It follows that E[X k ] < c/(1 + k) for all k ≥ 0.

Convergence to Poisson of iterated recombination
The purpose of the shattering condition is to guarantee that recombination bring the population distribution of genotypes back to that of a Poisson random measure, even in the face of the countervailing force of selection. However, before we proceed to our main result in which recombination, mutation, and selection are intermingled, it is instructive to see how recombination does its job in the absence of selection. This section shows that, under the shattering condition, the recombination process acting alone rapidly shuffles the distribution of a non-Poisson random measure on M to produce a probability measure on G that is close to the distribution of a Poisson random measure on M.
Along with the concept of annealed recombination, our result on convergence of iterated recombination in the absence of mutation and selection also requires the following condition on the initial distribution of genotypes in the population.
We cannot expect iterated recombination to to send an arbitrary probability measure Q on G to Poisson. In the extreme situation where the intensity measure µQ is diffuse and yet a realization of Q is not simple, recombination is unable to break up atoms of size greater than one, and so no amount of recombination will produce a result that is close to a Poisson random measure with intensity µQ. We have, of course, adopted the standing assumption that all of our probability measures on G are the distributions of simple random measures, but it is clear that if Q is, in some sense, close to the distribution of a random measure that is not simple, then the convergence of R k Q to the Poisson measure Π µQ can at best be arbitrarily slow. The following condition quantifies how far a probability measure Q is from one with a realization that has atoms and it will enable us to provide a bound on the rate that R k Q converges to Π µQ .
Definition 5.8. A probability measure Q on G is dispersive if there is a constant β such that for any Borel set A ⊆ M, Of course, the distribution of a Poisson random measure is always dispersive.
Theorem 5.9. Suppose that the pair (R, µQ) is shattering and Q is dispersive with constant β. Then, R k Q converges to the Poisson measure Π µQ as k → ∞, with Proof. Recall that for a measure g ∈ G and a Borel set A ∈ B(M) that g A ∈ G is the measure g(· ∩ A). Write Q A for the push-forward of the probability measure Q by the map g → g A . (This notation is used only within the context of this proof, and is not to be confused with the use of | elsewhere to denote the restriction of a measure to a set.) Let A = {A 1 , . . . , A L } be a partition of M. Then, by definition, the pushforward of R A Q by the map g → (g A1 , . . . , g A L ) is Q A1 ⊗ · · · ⊗ Q A L . Similarly, the push-forward of Π µQ by the map g → (g A1 , . . . , g A L ) is Π µQ A1 ⊗ · · ·⊗Π µQ A L . Suppose that F : G → R is Lipschitz with Lipschitz norm F Lip at most 1. Define Φ : G L → R by Φ(g 1 , . . . , g L ) = F (g 1 +· · ·+g L ). Note for fixed g 1 , . . . , g i−1 , g i+1 , . . . , g L that the function g → Φ(g 1 , . . . , g i−1 , g, g i+1 , · · · , g L ) is Lipschitz with Lipschitz norm at most 1 and so, writing for the product of measures, Therefore, Given a Borel subset A ⊆ M, defineπ to be the measure on M given bŷ Clearlyπ ≤ µQ. Also, define the sub-probability measureQ on G bŷ Observe thatπ is the intensity measure ofQ. Note also that We have We bound the first term on the right by For a bound on the second term on the right of (5.6), for any F : G → R with F Lip ≤ 1, we can writeπ[F (δ · )] for F (δ m ) dπ(m), and we have Thus, Finally, for our bound on the third term of (5.6), we have, by Lemma A.12, Thus, Putting (5.7), (5.8) and (5.9) into (5.6), we get (5.10) By (5.4), then, It follows, via Jensen's Inequality, that Applying Lemma 5.7 completes the proof of (5.3).
Note that Theorem 5.9 is essentially a random measure version of Le Cam's Poisson convergence result of [LC60].
Theorem 5.9 is about the situation where the initial distribution of genotypes in the population satisfies both a shattering condition and a dispersive condition and recombination acts alone. Our main convergence theorem in Chapter 8 posits an initial distribution of genotypes that satisfies a shattering condition along with an hypothesis on its Radon-Nikodym derivative with respect to the distribution of the Poisson measure with the same intensity measure, but no dispersive condition. When the selective cost S and the mutation rate ν are set to zero, the main convergence theorem also covers the situation in which recombination acts alone. The relationship between these two results on iterated recombination is clarified by the following observation, where we recall the definition of the quantity Θ(P, Q) from (2.7).
Proposition 5.10. Suppose that P = Π π and Q is a probability measure on G such that µQ = π and Θ(P, Q) < ∞. Then, there exists a positive constant β for which the dispersive condition Proof. SetĤ := Θ(P, Q). Lemma A.13, to be proved in the appendix, shows that the Lipschitz condition implies that Splitting M up into two sets A and A and using the independence properties of the Poisson probability measure P , we can bound this quantity by a constant multiple of ≤ e −π(A) eĤ π(A)eĤ π(A) exp{eĤ π(A)}.
The last expression is bounded by a constant multiple of π(A) 2 , so the dispersive condition holds.

Atoms in the initial intensity
We have discussed the shattering condition in the context of the mutation measure ν. We shall also be assuming that the initial intensity measure ρ 0 for the starting state P 0 is shattering. This hypothesis excludes initial intensity measures with atoms. The question arises as to whether atoms could be allowed. The answer is no.
The reason for excluding atoms is illustrated by an artificial case in which the mutation measure ν is zero and ρ 0 has a single atom at a point m ∈ M. As time passes m remains the only mutation present in the population, since recombination does not introduce mutations not already present. Each occurring element of G consists of an integer mass at m. We identify G with the nonnegative integers N 0 and simply write g for g(m). We can then think of S as a function from N 0 to R + . Take the initial probability measure Q 0 to be the distribution of a Poisson random variable with mean λ = ρ 0 (m). As usual, the notation t denotes the greatest integer less than or equal to the real number t.
After tn generations of selection, in the notation of Notation 4.4, the population is described by a measure Q which, by (4.2), equals .
This mass converges as n → ∞ to which, in our case, is a probability measure on the nonnegative integers that assigns to each integer g the mass This probability measure is not Poisson except in the non-epistatic case case where S(g) = cg for some constant c > 0. In short, in the presence of an atom in the initial intensity, the operation of selection drives the distribution of genotypes in the population away from that of a Poisson random measure, and recombination is powerless to restore its Poisson character. For this reason, something akin to the shattering condition for the recombination measure and the initial intensity is essential.

Preview of the main convergence result
We can now state the hypotheses and conclusions of our main result, the result that justifies the continuous-time model (2.10) as a limit of discrete-time models in which recombination acts on a faster time scale than mutation and selection. The formal statement of the theorem and its proof are given in Chapter 8, Theorem 8.1. The proof is based on supporting results established in Chapters 6 and 7. The statement of the theorem is given here by way of preview, so as to make the purposes served by the supporting theorems and lemmas in those chapters easier to discern.
We continue to use the notation x to denote the greatest integer less than or equal to the real number x.

Hypotheses:
Let (ρ t ) t≥0 be the measure-valued dynamical system of (2.10) whose existence is guaranteed by Theorem 2.4. Suppose that the selective cost function S satisfies the hypotheses of Theorem 2.4, namely • S(0) = 0, • S(g) ≤ S(g + h) for all g, h ∈ G, • for some constant σ, S(g) − S(h) ≤ σ g − h Was , for all g, h ∈ G.
In addition, suppose that the following assumptions are in force.
• The pair (R, ν) consisting of the recombination measure and the mutation measure is shattering. • The pair (R, ρ 0 ) consisting of the recombination measure and the initial intensity is shattering. • The initial measure Q 0 is equivalent to its Poissonization P 0 := PQ 0 = Π µQ0 = Π ρ0 , and Θ(P 0 , Q 0 ) < ∞.

Conclusions:
Then, for any T > > 0, If, in addition, the initial measure Q 0 = P 0 is Poisson, then this equation holds for = 0.
Remark 5.11. We have assumed throughout, for notational convenience, a particular order of operations -in each generation there is first selection, then mutation, then recombination. This order has no special significance, and the proofs hold equally well for another order, or if the same total amounts of mutation and selection were split up into multiple bouts within a generation, whether before or after recombination.
Remark 5.12. Because of Lemma 5.4, the hypotheses of Theorem 8.1 imply that the mutation intensity measure ν and the initial intensity ρ 0 are both diffuse. It follows easily from this that each probability measure Q k assigns all of its mass to the set of elements of G that have atoms of mass one; that is, every Q k is the distribution of a simple integer-valued random measure.
From now on, we assume without further comment that all the probability measures on G we consider are distributions of simple integervalued random measures.
Before beginning the formal presentation in the following chapters, we outline the strategy of the proof. The distance in the Wasserstein metric between our continuous-time dynamical system and our discrete-generation system at any point in time for any given value of the generation-scaling parameter n can be broken into three pieces: (1) the distance between the dynamical system and (with a suitable re-scaling of time) the discrete-generation system O k with complete Poissonization after every generation; (2) the distance between O k and the discrete-generation system with Poissonization only at the end of k steps, PQ k , which we may call end-state Poissonization; (3) the distance between PQ k and the discrete-generation system Q k itself. Chapter 6 is devoted to showing that the first distance, the one between Π ρt and O kt , is of order 1/n. This part of the proof is the only one in which the dynamical system comes into play, and it is also the only one in which recombination does not have a role because both systems are built (implicitly for the first and explicitly for the second) with complete Poissonization rather than a finite amount of recombination. Here we must focus primarily on understanding the operation of selection when the selective cost is scaled by 1/n.
Chapter 7 lays the groundwork for bounds on the second distance and the third distance. The second distance is a distance between the distributions of two Poisson random measures, which, as we know from Lemma A.12, can be bounded in terms of the distance between their intensity measures. The probability measure PQ k has the same intensity measure as Q k , so the quantity to be bounded is the distance between µO k and µQ k . We show in Chapter 8 that such a bound can be derived from a bound on the third distance, that between the probability measures PQ k and Q k .
The required bound on the third distance has to hold for every generation between zero and T n . The approach in the proof is to pick any intermediate generation as a provisional starting point and look at the dynamical systems after k steps beyond such a starting point. The dynamical systems differ by some terms that involve the initial states and some terms which primarily involve the selection costs accumulated over k steps. Chapter 7 is largely devoted to showing that recombination drives down the difference terms involving the starting states by a factor of k + 1, while terms involving selection costs remain bounded by a constant multiple of k/n. It follows that taking k = √ n steps beyond any starting point makes the former kind of terms drop by 1/ √ n before the latter kind of terms grow by more than 1/ √ n. The dynamical system are getting closer to each other over a time interval of length k/n = 1/ √ n which becomes infinitesimal as n goes to infinity.
In our infinite-population setting, as we have said, the process of selection is a process which reweights the probabilities of finding various combinations of mutant alleles in the genotype of a randomly selected individual. The tree structure of recombination events can be specified prior to the introduction of selection. Because the transformation of probability measures by reweighting is at issue, it makes sense to represent the probability measures Q k and O k by their Radon-Nikodym derivatives with respect to a suitable underlying probability measure. The Poisson probability measure P k serves this purpose. These probability measures, and their starred counterparts that keep track of the vintages of mutations from wild type, are separate objects for each separate generation. Joint distributions of variables across generations and sample paths of time-serial processes are not in the foreground. However, in the background lie the tree-structure of recombination and the accumulating effects of selection which do reach across generations and do lead to such joint distributions. The decomposition of genotypes in terms of vintages allows the relevant features of these background joint distributions to be inferred from current states, and so facilitates the proof.

Convergence with complete Poissonization
We embark on the proof of our main convergence result, Theorem 8.1, by establishing in this chapter a result, important in its own right, which provides the first step in that proof. For this result, Theorem 6.3, the recombination operator R is replaced by the operator P defined in (4.3) that immediately replaces a probability measure on G by the distribution of a Poisson random measure with the same intensity measure. In place of Q k := (RM n S n ) k Q 0 , we study the sequence of probability measures O k := (PM n S n ) k P 0 , defined in Notation 4.4. Both the measures O k and the measures Π ρt that define our continuous-time dynamical system are Poisson measures, determined by their intensity measures. It is simpler to work with the intensity measures, defined on M, than with probability measures defined on G. The relationship between discrete generations and continuous time, mediated by the scaling parameter n, has to figure prominently in the proof. Theorem 6.3 is essentially a shadowing result, but none of the standard shadowing theorems, such as those in [CKP95], seem to cover this case.
Remark 6.1. In this and the following chapters we frequently make use of multiple "constants", numbered successively as c 1 , c 2 , . . . or C 1 , C 2 , . . . . These constants may depend on on the time horizon T , the Lipschitz constant σ for the selective cost S, the total mass ν(M) of the mutation measure, and the total mass ρ 0 (M) of the intensity measure for the initial population.
Notation 6.2. Write π k := µO k for the intensity measure of the Poisson probability measure O k . Theorem 6.3. There are constants A and B, depending on σ, ν(M) and ρ 0 (M), such that for every T > 0, Proof. It suffices by Lemma A.12 to establish the first inequality. Note that sup 0≤t≤T π tn − ρ t Was ≤ sup 0≤k≤ T n π k − ρ k/n Was + sup k/n≤t≤(k+1)/n∧T ρ t − ρ k/n Was . (6.1) We obtain a bound on the first term on the right-hand side of (6.1), namely, (6.2) sup 0≤k≤ T n π k − ρ k/n Was by establishing a bound for π k+1 − ρ (k+1)/n Was in terms of π k − ρ k/n Was and then iterating that bound. Given π ∈ H + and the associated Poisson random measure X π with intensity π, define H π : M → R + by By the assumptions on the selective cost S and the inequality where d here is the metric on M. We take advantage of the Lipschitz constant σ for the selective cost S to write 0 ≤ S(g) ≤ σg(M), which implies for any π ∈ H + that Thus, for any π , π ∈ H + , and Now we may apply Lemma A.11 with β = 0 and C = σ/n, followed by Lemma A.12, to obtain Now, returning to the task of bounding the quantity in (6.2), Was + H π k · π k − H ρ k/n · π k Was + H ρ k/n · π k − H ρ k/n · ρ k/n Was (6.7) Consider the first term on the right-hand side of (6.7). By definition, Thus, Hence, for a bounded Borel function f : where we have again made use of Campbell's Theorem (see Proposition A.8). Equivalently, and so the first term on the right-hand side of (6.7) is zero. It follows from (6.8) and (6.3) that (6.9) π k (M) ≤ ρ 0 (M) + k n ν(M), k ≥ 0.
For the last term on the right-hand side of (6.7) we have Was + 1 n 0 F ρ s+k/n · ρ s+k/n − F ρ k/n · ρ k/n Was ds. (6.12) By Lemma A.2 we know for any ρ ∈ H + that and a similar bound holds with X ρ replaced by X ρ + δ x . Note also that Applying the obvious bound (6.13) ρ t (M) ≤ ρ 0 (M) + tν(M), t ≥ 0 and the inequality 1 − y ≤ e −y ≤ 1 − y + y 2 2 (for y ≥ 0), it follows that (6.14) Thus, for a suitable constant C 3 . From Lemma 2.5 and (6.13), for any 0 ≤ u ≤ v, (6.16) In particular, for 0 ≤ s ≤ 1 n , F ρ s+k/n · ρ s+k/n − F ρ k/n · ρ k/n Was ≤ F ρ s+k/n · ρ s+k/n − F ρ s+k/n · ρ k/n Was where we combined (6.16) and (6.13) with Lemma 2.5 and Lemma 2.6 in the last inequality. Hence, for a suitable constant C 4 . Combining (6.15) and (6.18), we see from (6.12) that for a suitable constant C 5 . Recall from (6.8) that the first term on the right-hand side of (6.7) is zero and bound the remaining terms using (6.10), (6.11) and (6.19) to obtain (6.20) π k+1 − ρ (k+1)/n Was ≤ 1 + C 6 n π k − ρ k/n Was + C 7 n 2 for (k + 1) ≤ nT , where C 6 and C 7 are constants. Iterating (6.20) leads to The result now follows by using (6.21) and (6.16) to bound the first and second terms, respectively, on the right-hand side of (6.1).

Estimates for Radon-Nikodym derivatives
In this chapter we establish a set of lemmas which provide the central technical tools used in the proof of the main convergence result in Chapter 8. As previewed in Chapter 5, we work with weights in the form of Radon-Nikodym derivatives, because selection, in our infinite population, is a process of reweighting of population frequencies of genotypes. Furthermore, we suppose that our initial population Q 0 can be treated as a reweighted version of the Poisson probability measure P 0 = PQ 0 . Thus, it is natural to express our full discrete-generation system in terms of Radon-Nikodym derivatives with respect to a family of Poisson probability measures.
The substantive effects of selection have been treated in Chapter 6. That chapter shows how, in the presence of selection, the completely Poissonized version of the discrete-generation system aligns with the continuous-time dynamical system. From this point on in the story, attention focuses on short stretches of time which become infinitesimal as the scaling parameter n goes to infinity. Over these short stretches, we show that the effect of selection can be treated as negligible whereas the total effect of recombination tracks closely that of complete Poissonization.
We thus require bounds on the overall effects of selection, acting along with mutation and recombination, over k generations. We are interested in the outcome at the k-th generation, but the effects of selection accumulate across generations and depend on the genotypes of the ancestors, portions of whose genomes are recombined into the genotype of any individual sampled from the population at generation k. Reconstruction of the history of accumulating selection is made possible by the decomposition of elements of G according to vintage and thus by the starred operators which update in each generation the population distribution of mutations from ancestral wild type labeled according to vintage.
The capstone of this chapter, Lemma 7.15 is a bound on the difference between the starred operator Q * k and its own Poissonization PQ * k . This bound is obtained by combining an approximation for dQ * k /dP * k derived in this section with an approximation for dPQ * k /dP * k derived in the following section. In both approximations, terms that are retained arise from contributions of the initial probability measures. Contributions from selection are absorbed into an error term bounded by a multiple of k/n.
We begin with an elementary observation concerning how the action of the starred recombination operator R * may be represented in terms of Radon-Nikodym derivatives.
Lemma 7.1. Suppose that P * is a Poisson random measure. Then, for g ∈ G k+1 , Proof. By Corollary A.7, for any bounded Borel function F : G k+1 → R, This is equivalent to the claim.
We next develop apparatus for reconstructing the ancestral genotypes that are consistent with a particular genotype (decomposed by vintages) that is observed at the end of k generations. At each ancestral generation, there is a partition of M (generally including empty sets), as presented in Section 4.3. Each set in this partition corresponds to an ancestor in the given generation. Mutant alleles within the set are retained from the genotype of the ancestor while mutant alleles outside the set are lost through recombination. We work with functions that assign to each set a surrogate for the missing piece of the ancestral genome. This additional bookkeeping facilitates our development of bounds on the "de-Poissonizing" effects of selection.
Definition 7.2. Let A be a partition of M into Borel sets and let ξ : A → G be a family of genotypes indexed by the sets of A. Given F : G → R and g ∈ G, set Note that if A is the trivial partition {M}, then F A (g, ξ) = F (g) for any ξ : A → G.
Lemma 7.3. Fix a Borel partition A of M and a family of genotypes ξ : A → G. Then, for any F : G → R, Proof. We have Notation 7.4. Recall the ancestral tree L and the random recombination sets R β and random genotypes Γ β indexed by the nodes β ∈ L introduced in Section 4.4. Fix a node ∈ L with vintage V ( ) = k. Any set A in the random partition A i ( ) = A k,i corresponds to a node ω with vintage V (ω) = i. Define ξ k,i : A k,i → G by ξ k,i (A) := Ξ ω = υ ω Γ υ | W (υ,ω) .
We now state and prove a lemma which gives an exact representation of the Radon-Nikodym derivative of the starred measure Q * k with respect to P * k at generation k in terms of ancestral episodes of selection. The factors that make the total probability integrate to 1 are absorbed into a normalization constant C.
Lemma 7.5. Set H := log dQ 0 /dP 0 . Suppose for k ∈ N 0 that the partitions A k,0 , . . . , A k,k are defined using the node ∈ L with vintage V ( ) = k. Then, where C does not depend on (g 0 , . . . , g k ) and the functions J i : G → R are given by J 0 (g) := H(g) − S(g)/n and J i (g) := −S(g)/n for 1 ≤ i ≤ k − 1. The expectation is taken with respect to the random collection of genotypes ξ k,i (or Ξ ) and the random partitions A k,i (or W (ω, )).
Proof. By definition, and The equality of the two expectations follows. The remainder of the proof is by induction on k. For k = 0, the partition A 0,0 is the trivial partition consisting of the single set M, and H A0,0 (g, ξ) = H(g), and so the result is obvious. Now suppose the result is true for k − 1 ≥ 0.
(7.1) by Lemma 7.1 and the observation that for any two probability measures Q * and P * on G k−1 we have Let the parents of the node be the two nodes , ∈ L with common vintage k − 1. Suppose first of all that the random partitions A k−1,0 , . . . , A k−1,k−1 are defined using the node . By the induction hypothesis, for some constant C 1 . Of course, the same equation holds if is replaced by .
Recall that R is a B(M)-valued random variable with distribution R and the collection of random sets By convention, the genotype of the individual coincides with that of the parent on the set R and with that of the parent on the complementary set R . For 0 ≤ i ≤ k − 1, define a random genotype by Note that the conditional distribution of Y := (Y 0 , . . . , Y k−1 ) given R is P * k−1 , so P * k−1 is also the unconditional distribution of Y and Y is independent of R . Thus, is the push-forward of the probability measure R ⊗ P * k−1 by the map which is in turn that of the push-forward of the probability measure R⊗P * k−1 ⊗P * k−1 by the map (A, (g 0 , . . . , g k−1 ), (g 0 , . . . , g k−1 )) → A, (g 0 , . . . , g k−1 )| A , A , (g 0 , . . . , g k−1 )| A .
Recall from (2.7) that for two probability measures P and Q we denote the total-variation Lipschitz constant of log dQ/dP by Θ(P, Q).
Proof. Consider first the claim for log dQ * k /dP * k . It suffices to check for any g ∈ G k+1 , x ∈ M, and 0 ≤ j ≤ k that where δ (j) x := (0, . . . , 0, δ x , 0, . . . , 0) ∈ G k+1 with δ x in the j th coordinate. Suppose first that j = 0. By Lemma 7.5, It follows from Lemma 7.3 that and (7.3) follows immediately. The proof for 1 ≤ j ≤ k is similar. In order to establish the claim for log dQ k /dP k , it suffices to check for any g ∈ G and x ∈ M that Define X to be the canonical random variable on G * . Observe that Under P * k , the conditional distribution of X given the event {ΣX = g = g(M) j=1 δ xj }, is the distribution of the G k+1 -valued random variable where the {0, 1, . . . , k}-valued random variables i j , 1 ≤ i ≤ g(M), are independent with distribution given by Consequently, for a bounded measurable function F : G k+1 → R, we have where i is a {0, 1, . . . , k}-valued random variable with distribution Thus, It is immediate from (7.3) that for 0 ≤ i ≤ k, and so (7.4) holds.
The proof of the inequalities for dQ * k /dP * k is similar.
Corollary 7.8. Suppose thatĤ := Θ(P 0 , Q 0 ) < ∞. Then, there are constants c and > 0 (depending onĤ) such that for any k ≤ n and any g ∈ G k+1 , Proof. Set H = dQ 0 /dP 0 . From Lemma 7.5 and Lemma 7.3, we know that there is a constant C (independent of g) such that, for the random map ξ k,0 : A k,0 → G introduced in Lemma 7.5, (Note: We include the expectation with respect to the random partition A k,0 in the E[·].) Substituting in the definition of H, we find that there is a constant C such that Since the random measures {ξ k,0 (A) : A ∈ A k,0 } are i.i.d. with common distribution P 0 , this becomes By Lemma A.13, We have for any set A ∈ B(M) that = exp ν(A)k n e σk/n − 1 · P 0 exp H(X) + σk n X(A) because g → (g 1 + · · · + g k )(A) is a Poisson random variable with mean k n ν(A) under P * k and g → g 0 is a Poisson random measure with the distribution P 0 under P * k .
Using the inequality e x − 1 ≤ xe x for x ≥ 0, we have exp ν(A)k n e σk/n − 1 ≤ exp σν(A)k 2 n 2 e σk/n ≤ exp where c 1 is a constant depending on σ and T and k/n ≤ . By the same inequality, the fact that P 0 [e H(X) ] = Q 0 [1] = 1, and Corollary A.14, where c 2 is a constant depending on σ,Ĥ, T , ν(M), and µP 0 (M) = ρ 0 (M). Thus, for any partition A, for a constant c 3 depending on σ,Ĥ, T , ν(M), and ρ 0 (M). On the other hand, using similar arguments and the inequality e −x ≥ 1 − x, x ≥ 0, we get for any A ∈ B(M) that where c 4 and c 5 are constants depending onĤ, T , ν(M), and ρ 0 (M). We may then find constants c 6 and > 0 such that for all k ≤ n, and so Combining the upper and lower bounds from above, we see that there is a constant c 7 such that Remark 7.9. The approximation to log dQ * k /dP * k in Corollary 7.8 involves a product over the (potentially very large) partition A k,0 and looks rather unwieldy. We may rewrite the approximation as follows to make it apparent where the nontrivial contributions to the product arise: (7.5) This is a trivial consequence of the fact that P 0 [e H(X) ] = 1.

Comparisons with complete Poissonization
Recall from Notation 4.1 that the Poissonization operator P acts on a probability measure P on G by PP := Π µP . Also, recall from Notation 4.4 that O k := (PS n M n ) k P 0 is the analogue of the sequence of probability measures Q k := (RS n M n ) k P 0 from Notation 4.4 that is of primary interest to us in Theorem 8.1, with the recombination operator R replaced by the complete Poissonization operator P. Finally, recall from Notation 4.13 that the counterpart of Q k for G * (that is, for the setting in which we keep track of the generation in which mutations from the ancestral wild type occurred) is Q * k = (R * S * n M * ) k P 0 . Lemma 7.10. Suppose that P = Π π for some finite measure π ∈ H + and Q is an equivalent measure. Put H := log dQ/dP . Set τ (m) := log G exp{H(g + δ m )} dP (g).
(a) The probability measure PQ has intensity measure µQ that is absolutely continuous with respect to π with Radon-Nikodym derivative The probability measure PQ is absolutely continuous with respect to P , with Radon-Nikodym derivative satisfying Proof. For a Borel function f : where we used Campbell's Theorem (see Proposition A.8) for the penultimate equality. Part (a) follows. It then follows immediately from Lemma A.4 that PQ is absolutely continuous with respect to P , with giving the first claim in part (b). The second claim in part (b) follows by another application of Campbell's Theorem.
(a) For all x ∈ M and k ∈ N 0 , (c) For all g ∈ G and k ∈ N 0 , log dO k dP k (g) ≤ kσ n g(M) + e kσ/n µP k (M) .
We now extend Lemma 7.10 to probability measures on G * . Recall that a (k+1)tuple of independent Poisson random measures on M with intensities π 0 , . . . , π k , may, as discussed in Section 4.4, be thought of as a single Poisson random measure on the set M × {0, . . . , k} with intensity π i ⊗ δ {i} .
Lemma 7.13. Suppose thatĤ := Θ(P 0 , Q 0 ) < ∞. Set H := log dQ 0 /dP 0 . Let τ (i) , 0 ≤ i ≤ k, be as in Corollary 7.12 with Q * = Q * k and P * = P * k . Then, there are positive constants and c, depending onĤ but not depending on k or n, such that Proof. We apply Corollary 7.8. Since the Poisson random measures ξ k,0 (A) are independent as A ranges over the random partition A k,0 , there are constants c 1 and such that for k ≤ n, where C does not depend on g, but may depend on k and n. We recombine terms to obtain Note that g → k i=1 g i (M) and g → g 0 are independent under P * k , with the former having a Poisson distribution with parameter kν(M)/n. A bound on the constant C can be imported from Lemma A.13, leading to the following inequality Applying Corollary A.7, we see that there is a constant c 2 such that where A(x) denotes the element of A k,0 containing x ∈ M. By Lemma A.15 there are constants 2 and c 3 such that for k/n ≤ 2 the product is bounded by Thus, there is a constant c 4 such that for k ≤ n, where = ∧ 2 , (7.6) τ (0) (x) ≤ c 4 k n + log E P 0 exp H(X + δ x ) + c 1 k n X(A(x)) .
We proceed by writing (7.7) Using again the relation e x − 1 ≤ |x|e |x| , and the fact that P 0 is Poisson, hence the restrictions of X to disjoint sets are independent under P 0 , the expectation in the numerator above may be rewritten as where c 5 := exp Ĥ + c 1 T . For the denominator, we have Combining these bounds and using the inequality log(1 + x) ≤ x for x ≥ 0, we may find a constant c 6 (depending on ρ 0 (M) andĤ) such that Combining this with (7.6) and (7.7) gives us a constant c 7 such that (7.8) for k/n ≤ . Nearly identical calculation give us a lower bound, completing the proof of the first claim. The second claim follows from similar, but simpler, arguments.
Corollary 7.14. Under the hypotheses of Lemma 7.13, Proof. By Corollary 7.12, the quantity on the left-hand side of the claimed inequality is at most We bound the quantities on each of the five lines of (7.9) separately. For the first line, the first part of Lemma 7.13 provides a bound of (ck/n)g 0 (M). For the second line, the second part of Lemma 7.13 provides a bound of (ck/n) k j=1 g j (M) For the third line, the same lemma lets us write By Campbell's Theorem A.8, the integral with respect to π 0 (dy) of the left-hand side of this last expression is π 0 (M) exp{−ck/n} and the corresponding integral on the right-hand side is π 0 (M) exp{+ck/n}. Subtracting π 0 (M) and making use once more of the inequality |e x − 1| ≤ |x|e |x| , we conclude that the quantity on the third line is bounded by π 0 (M)(ck/n)e c when k/n ≤ .
For the fourth line, again by Campbell's Theorem, the integral is the sum of the total intensities of g 1 to g k with respect to Q * k , that is to say, By Corollary 7.7, this nonnegative quantity is bounded above by (g 1 (M) + · · · + g k (M)) exp(cg 0 (M) + · · · + cg k (M) + cµP k (M))) dP * k (g).
Under P * k , the random measures g → g 0 and g → g 1 + . . . g k are independent Poisson random measures with respective intensity measures ρ 0 and kν/n, and so Proof. Combining Corollary 7.8 (in the form of (7.5)), with Corollary 7.14, we see that there are positive constants and c 1 , depending onĤ, such that for k/n ≤ , where |γ(g)| ≤ c 1 and is a quantity not depending on g.
Observe that the product in the second term of (7.10) is universally bounded above and below as follows We decompose the product as (7.12) The logarithm of the product in the first expectation on the right is   log (7.13) We next find bounds for the second and third terms on the right of (7.13). For any set A ∈ B(M), whereȞ :=ĤeĤ . A similar calculation gives an analogous lower bound for the second term on the right of (7.13), yielding The third term on the right of (7.13) may be bounded in the same way. Thus, Equivalently, where R(g, A k,0 ) is a random variable satisfying We see from Lemma A.3 that Combining this with (7.11) and (7.12) we may find constants c 2 , c 3 (depending onĤ and ρ 0 (M)) such that We may now apply part (b) of Lemma A.13 with and f 1 (g) = 0, using the fact that f 2 (g) is uniformly bounded by a constant c 4 , to obtain By Corollary 7.7 there are positive constants c 5 and c 6 such that dQ * k dP * k (g) ≤ c 6 e c5Σg(M) . Thus, for a constant c 7 . Note that Σg = g 0 + (g 1 + · · · + g k ) and g → g 1 + · · · + g k is independent of of g → g 0 under P * k , and so the resulting integral is simply included in c 7 .
We may bound the indicator of the event { A g (A) ≥ 2} from above by the sum A∈A k,0 1{g (A) ≥ 2}. Exchanging the order of the integrals in the second and third terms, we then obtain This allows us to conclude for some constant c 8 that We apply Lemma 5.7 to the second term on the right-hand side and find a constant c 9 , which now depends on the shattering constant α, such that We also have P k [X(M)] equal to (ρ 0 (M) + kν(M)/n), which is bounded by a constant for k/n ≤ T . It follows that completing the proof.

CHAPTER 8
Convergence of the discrete-generation system 8.1. Outline of the proof We have now assembled all the ingredients for our proof of our main convergence result previewed in Chapter 5. This result establishes the convergence of our discrete-generation dynamical system to our continuous-time dynamical system as the scaling parameter n goes to infinity. This parameter n governs the rate of recombination relative to rates of selection and mutation. The overall strategy behind the proof has been described at the end of Chapter 5. Here we outline some more detailed considerations that enter into the proof.
Thanks to the result on complete Poissonization proved in Chapter 6, the task that remains involves only two Wasserstein distances which must be bounded uniformly with bounds that go to zero as n goes to infinity. One is the distance within the Poisson family between the probability measures O k and PQ k . The other is the distance between the Poisson family and the discrete-generation system Q k itself, gauged by the distance between Q k and its own Poissonizaton PQ k .
The distance within the Poisson family can be bounded by the Wasserstein distance between the corresponding intensity measures. A critical observation, encapsulated in the inequality (8.3), is that the intensity measures deviate from each other appreciably only when Q k deviates from Poisson. This reduces the problem to bounding the distance from the Poisson family, the distance between Q k and PQ k .
This problem is addressed in two parts, by parts requiring results established in Chapter 7. First we adduce the formula for the Radon-Nikodym derivative of Q k with respect to P k in Lemma 7.5. This formula is exact, in principle, but is probably impossible to compute exactly in most cases. Next we rely on two approximations derived in subsequent lemmas.
• Uniformly on k ≤ T n, there is a bound on Θ(P 0 , Q 0 ). • For k n, there is a bound on PQ k −Q k Was of the form c(1/k)∨(k/n), where the constant c depends on Θ(P 0 , Q 0 ).
We bring these consequences to bear by noting that there is no reason why we need to start at the original P 0 and Q 0 . Since Θ(P L , Q L ) is bounded for any L ≤ T n, we have a bound PQ L+ n for any L. This approach covers all k in the desired range, except in a small neighborhood of 0. If the initial probability measure Q 0 is Poisson (that is, Q 0 = P 0 ), then we have a bound PQ k − Q k Was ≤ ck/n for small k, and this covers the gap at 0. If Q 0 is not Poisson, then there is a failure of uniform convergence in a neighborhood of 0: as n → ∞, the non-Poisson probability measure at time 0 jumps immediately to its Poissonization at time 0+.

The convergence theorem
Theorem 8.1. Let (ρ t ) t≥0 be the measure-valued dynamical system of (2.10) whose existence is guaranteed by Theorem 2.4. Suppose that the selective cost function S satisfies the hypotheses of Theorem 2.4, namely • S(0) = 0, • S(g) ≤ S(g + h) for all g, h ∈ G, • for some constant σ, S(g) − S(h) ≤ σ g − h Was , for all g, h ∈ G. In addition, suppose that the following assumptions are in force.
• The pair (R, ν) consisting of the recombination measure and the mutation measure is shattering. • The pair (R, ρ 0 ) consisting of the recombination measure and the initial intensity is shattering. • The initial measure Q 0 is equivalent to its Poissonization P 0 := PQ 0 = Π ρ0 , and Θ(P 0 , Q 0 ) < ∞. Then, for any T > > 0, If, in addition, the initial measure Q 0 = P 0 is Poisson, then this equation holds for = 0.
Proof. The proof proceeds by establishing a bound for the total distance Π ρt − Q k Was , a bound uniform across generations k in the set {k ∈ N 0 : n ≤ k ≤ T n } which goes to zero as the scaling parameter n goes to infinity. By the triangle inequality we have The first of the three distances on the right-hand side has already been shown in Theorem 6.3 to be bounded by 1/n times a constant depending on T . In other words, the measure describing the dynamical system is close to the measure O k = (PM n S n ) k Q 0 , the Poisson probability measure that describes a population after k rounds of selection, mutation, and complete Poissonization.
What remains is to bound the second and third of the distance terms. We need to show that the Poisson measure O k is close to the Poisson measure PQ k , and that PQ k is close to Q k itself, where Q k = (RM n S n ) k Q 0 is the probability measure that describes a population after k rounds of selection, mutation, and recombination.
By Lemma A.12 the second term on the right in (8.1) is bounded by a multiple of the Wasserstein distance between the intensities µO k and µQ k (the latter of which is the same as µPQ k ): We introduce symbols for the two terms for which we are now seeking bounds. Set a k := PQ k − Q k Was and b k ; = µQ k − µO k Was .
We treat b k first, eventually obtaining a recursion equation for it that turns out to involve a k .
Given a Borel function f : M → [−1, 1] with f Lip ≤ 1, define the function F : G → R by F (g) := g[f ] = f (m) dg(m). The Lipschitz condition on f implies By definition, the integral µQ[f ] is the expected value Q[F ] for any probability measure Q.
By definition of the family (Q k ) k≥0 , The recombination operator R leaves intensities unchanged, and the mutation operator M n adds the measure ν/n to the intensity of any probability measure on G. Thus, The analogous expression holds for µO k+1 and O k . By definition of the Wasserstein metric, Use of this equation requires an expression for the difference S n Q k [F ] − S n O k [F ] . For any Borel function F : G → R, The product of the two reciprocal factors on the right-hand side of (8.4) is bounded, via Jensen's Inequality, by exp((Q k [S]+O k [S])/n). The remaining factor, with all its terms, is easier to handle if we rewrite it in terms of expected values of 1 − exp{−S(g)/n} and F (g)(1 − exp{−S(g)/n}), quantities that are small when n is large.
Set H(g) := F (g)(1 − exp{−S(g)/n}). Then, the final factor in (8.4) equals , and PQ k [exp{−S/n}] ≤ 1. We may then bound (8.4) by (8.5) Using Lemma 7.6 and Lemma 7.11, we may find positive constants c 1 , c 2 such that If we write c 3 := c 1 (σ ∨ 1)µP T n (M)e (e c 2 −1)µP T n (M) , then (8.7) As F (g) = g[f ], Observe that |H(g)| ≤ σg(M) 2 /n and for g, g ∈ G, We may then apply Lemma A.11 with β = 1 and C = 4σ/n to obtain A simpler version of this computation completes the bound of the third line in (8.5) by for an appropriate constant c 4 . We would like to bound the remaining terms with respect to the Wasserstein distances among the measures involved, but this is not possible directly, because the function H is neither bounded nor Lipschitz. To get around this problem we truncate F and S, and then make use of the Poisson bounds that we have on the probability of large values of these functions. Fix some positive M , and define The function n(3M 2 σ) −1 H (M ) is bounded by 1/3 and has Lipschitz constant bounded by 2/3, so n(3M 2 σ) −1 H (M ) Lip ≤ 1. Then in the penultimate inequality. A similar (but simpler) calculation together with Stirling's formula shows that we may find constants c 5 and c 6 such that the fourth line of (8.5) is bounded by (8.10) Taking M = log n and combining this with (8.3), (8.8), (8.9) and (8.10), we may find constants c 7 , c 8 , c 9 such that (8.11) b k+1 ≤ e c9/n b k + c 7 log 2 n n a k + c 8 n log log n .
We now take the important step mentioned in Section 8.1 of considering intermediate states of the systems (Q k ) k≥0 and (O k ) k≥0 as starting states for the purposes of applying the bounds from Chapter 7 over time intervals that are short in the scaling limit. The starting state Q 0 from which (Q k ) k≥0 and (O k ) k≥0 are derived can be quite general, although there are a few conditions that must be satisfied for the bounds in Chapter 7 to apply. If an intermediate state Q L can be shown to satisfy these conditions, then we can treat Q L as a new starting state, count generations forward from L, and apply the results of Chapter 7. Recall that some of the bounds in these lemmas go down like 1/(k + 1) and some of them go up like k/n. The best bounds are achieved by going forward on the order of k = √ n steps from an intermediate state at generation L with 0 ≤ L ≤ T n − k. The result will be a bound on a L+k . Define Q k := Q L+k , P k := P L+k , Q * k := (R * M * n S * n ) k Q L , P * k := (M * n ) k P L .
In order to apply the lemmas of Chapter 7, it is necessary to check a Lipschitz condition and a shattering condition. With regard to the Lipschitz condition, Lemma 7.6 guarantees the constant upper bound Θ( P 0 , Q 0 ) ≤ Θ(P 0 , Q 0 ) + T σ.
With regard to the shattering condition, it is necessary for the recombination measure R to be shattering with respect to the initial intensity, which for Q is the measure µQ L .
We verify the shattering condition by an argument analogous to the proof of Lemma 5.4. Set π L = µP L and choose any Borel set A of M. According to the upper bound on Radon-Nikodym derivatives in Corollary 7.7, there is a positive constant c such that µQ L (A) = g(A) dQ L dP L (g) dP L (g) ≤ g(A) exp{cg(M) + cπ L (M)} dP L (g).
Thus, µQ L (A) is bounded above by π L (A) times a constant depending on π L (M) but not on A. Similarly the lower bound on Radon-Nikodym derivatives in Corollary 7.7 provides lower bounds on µQ L (A∩R) and µQ L (A∩R ) for any segregating set R, bounds which are multiples of π L (A ∩ R) and π L (A ∩ R ). Because π 0 is assumed to be shattering with respect to R, so is π L by Lemma 5.4. Hence, we can find a new shattering constant α such that and µQ L is shattering. A bound can now be established on α L+k . Recall that P * k is the Poisson measure on G * obtained by starting with the Poisson measure P 0 and solely applying the mutation operator M * n (without selection or recombination) to the distribution of genotypes broken down by vintages.
It follows from the relations Σ Q * k = Q k , ΣP * = PΣ, and Σ P * k = P k that ≤ c 9 1 k + 1 ∨ k n for some constant c 9 , by Lemma 7.15. Setting k = √ n and combining this with the bound (8.11), we obtain for all nT ≥ j ≥ √ n, b j+1 ≤ e c6/n b j + c 7 log 2 n n a j + c 8 n − log log n a j ≤ c 9 n −1/2 , (8.12) Thus, there are constants c 6 and c 10 such that (8.13) b j+1 ≤ e c6/n b j + c 10 log 2 n n 3/2 .
For 0 ≤ j ≤ √ n, b 0 = 0, b j+1 ≤ e c6/n b j + c 7 log 2 n n a j + c 8 n − log log n , a j ≤ c 9 (j + 1) −1 , (8.14) implying that b j ≤ c 11 log 3 n/n for some constant c 11 . Taking this bound on b √ n as a starting point for iterating (8.13) we conclude that there is a constant c 12 such that for nT ≥ j ≥ √ n, b j ≤ c 12 log 2 n n 1/2 . Returning to (8.2), this implies that Q k − O k Was ≤ c log 2 n n 1/2 . Combined with Theorem 6.3, this immediately implies lim n→∞ sup ≤t≤T Π ρt − Q tn Was = 0 for > 0.
When Q 0 is already Poisson, we begin with a 0 = 0, allowing us to extend (8.12) to k in the range 0 ≤ k < √ n, completing the proof.

A.1. Gronwall's Inequality
We make frequent use of the following version of the inequality discovered by Thomas Hakon Gronwall in 1919 and extended by Richard Bellman in 1943, so we include a statement for easy reference. A proof may be found in Appendix 5.1 of [EK86].
Proposition A.1. Let f be a nonnegative Borel measurable function on R + that is bounded on bounded intervals. Suppose there exists a constant L > 0 such that for all t ≥ 0 Then, f (t) = 0 for all t.

A.2. Two expectation approximations
The following two lemmas give give error bounds for approximations used several times in this work. In particular, if Y is bounded by a constant τ , then Proof. Jensen's Inequality applied to the convex function y → exp{−y} im- For nonnegative y, the function y → (1/2)y 2 −exp{−y} is also convex. Jensen's Inequality implies Taking logarithms of both sides and using the bound log(1 + x) ≤ x, x ≥ −1, complete the proof.
Lemma A.3. Let Y be a random variable with finite first moment. Then, Proof. The first inequality is immediate from Jensen's inequality and the convexity of y → exp{y}. For the second inequality, use the inequalities log(1+y) ≤ y, y ≥ −1, and exp{x} − 1 ≤ x exp{x}, x ≥ 0, to get

A.3. Identities for Poisson random measures
The following elementary lemma is well-known and follows readily from the fact that the conditional distribution of the Poisson random measure X π given the event {X π (M) = n} is the distribution of the random measure n i=1 δ Zi , where Z 1 , . . . , Z n are i.i.d. M-valued random variables with common distribution π/π(M).
Lemma A.4. Suppose that π , π ∈ H + and π is absolutely continuous with respect to π . Then the Poisson probability measure Π π is absolutely continuous with respect to Π π with Radon-Nikodym derivative F (g A + g A )H (g )H (g ) dΠ π (g )dΠ π (g ) Proof. Note that the vector of random measures (X A , X A , X A , X A ) has the same distribution as (Y A , Y A , Y A , Y A ). Then observe that An important property of Poisson random measures, which we use in our discussion of the starred probability measures, is that we can construct a Poisson random measure on a product space by starting with a Poisson random measure that lays down point on one component, and then conditional on that scatter of points choosing the corresponding second coordinates independently according to some Markov kernel. One clear statement of this property is Theorem 6.3.2 of [Ç ın11]. A version tailored to our particular application is the following.
For the sake of completeness, we record the following result that follows from combining Lemma A.5 and Proposition A.6. Recall that for g = (g 0 , . . . , g k ) ∈ G k+1 and A ∈ B(M) we define g A := (g 0 ) A , . . . , (g k ) A . We can identify g with the measure on M × {0, . . . , k} that assigns mass g i (B) go the set B × {i}, and g A can be identified in the same way with the restriction of that measure to the set A × {0, . . . , k}.
Corollary A.7. Let P be the distribution of a Poisson random measure on M × {0, 1, . . . , k}. Fix a Borel set A ⊆ M. Identifying P with a probability measure on G k+1 , we have for bounded Borel functions F, H , H : G k+1 → R that G k+1 G k+1 F (g A + g A )H (g )H (g ) dP (g )dP (g ) The following version of Campbell's Theorem is one of a cluster of results relating to Palm probabilities inspired by 1909 work of N. R. Campbell. It follows immediately from two results proved in [DVJ07], Proposition 13.1.IV (an equality involving general Palm kernels) and Example 13.1c (an expression for the Palm kernel of a Poisson Process). (π(M)e c ) j j! j! (j + s)! j r .
The next result says that if the intensities of two Poisson random measures are close in the Wasserstein sense, then the same is true of their distributions. In the statement of the result, the Wasserstein distance on the left-hand side of the inequality is between probability measures on the space G, while the Wasserstein distance on the right-hand side is between finite measures on M.
In particular, if we first build X π and then construct X π * by the usual "thinning" procedure of independently keeping each point of X π with probability π (M)/π (M) and discarding it with the complementary probability, we have where we used the inequality a k − b k ≤ ka k−1 (a − b) for a ≥ b ≥ 0. Note that if Y is an integer-valued Poisson random variable with mean λ, then writing S(β, k) for the number of partitions of a set of size β into k blocks (that is, the Stirling number of the second kind) and B β = β k=0 S(β, k) for the number of partitions of a set with β members (that is, the Bell number) E[Y β ] = β k=0 S(β, k)λ k ≤ B β (λ β ∨ 1) ≤ β β (λ β ∨ 1), because B β ≤ β β (any partition of a set with β elements that has k blocks defines β(β − 1) · · · (β − k + 1) different functions from the set into itself by assigning a distinct element of the set to be the image of all the elements in each block, and there are β β functions from the set into itself). Therefore, Thus, Π π [H] − Π π * [H] ≤ 2C(2β) β (π (M) ∨ 1) β π (M) − π (M) ≤ 2C((2β)(π (M) ∨ 1)) β π − π Was .
The next result, is a special case of Lemma A.11. It probably already exists in some form in the literature, but we have been unable to find a reference.
A.5. Bounds for Radon-Nikodym derivatives Lemma A.13. Let Q and Q be equivalent probability measures on some measurable space X . Suppose there are functions and f 1 , f 2 : X → R such that C exp{f 1 (x) − f 2 (x)} ≤ dQ dQ (x) ≤ C exp{f 1 (x) + f 2 (x)}.
(b) Moreover, if f 1 , f 2 : X → R + , then Proof. We have and (a) follows. Now, By Jensen's Inequality, Note also that if a, b ≥ 1, then | a b − 1| ≤ ab − 1; this is obvious when a ≥ b, and when a < b the inequality is equivalent to ab 2 − 2b + a = a(b − 1) 2 + 2(a − 1)b > 0. Thus, where we have used the inequality e y −1 ≤ ye y for y ≥ 0, and the result follows.
Corollary A.14. Set Q = Π π for some π ∈ H + and suppose that the probability measure Q is equivalent to Q withĤ := Θ(Q, Q ) < ∞.   Lemma A.15. Suppose that Q = Π π for some π ∈ H + and the probability measure Q is equivalent to Q withĤ := Θ(Q, Q ) < ∞. We now proceed to estimate We can then estimate the ratio exp{H(g) + βg(A)} dQ(g) We now proceed to estimate Thus, there is a positive constant C (depending onĤ and π(M)) such that R ≤ 1 + Cβπ(A) 1 − Cβ π(A) .
Part (b) follows immediately. Proof. Since e α ≥ 1, for any nonnegative function F :