Exponential growth rates in a typed branching diffusion
Exponential growth rates in a typed branching diffusion
The Annals of Applied Probability
2007, Vol. 17, No. 2, 609–653
DOI: 10.1214/105051606000000853
c© Institute of Mathematical Statistics, 2007
EXPONENTIAL GROWTH RATES IN A TYPED
BRANCHING DIFFUSION
By Y. Git, J. W. Harris1 and S. C. Harris
Cambridge University, University of Bristol and University of Bath
We study the high temperature phase of a family of typed branch-
ing diffusions initially studied in [Astérisque 236 (1996) 133–154] and
[Lecture Notes in Math. 1729 (2000) 239–256 Springer, Berlin]. The
primary aim is to establish some almost-sure limit results for the long-
term behavior of this particle system, namely the speed at which the
population of particles colonizes both space and type dimensions, as
well as the rate at which the population grows within this asymp-
totic shape. Our approach will include identification of an explicit
two-phase mechanism by which particles can build up in sufficient
numbers with spatial positions near −γt and type positions near
t at large times t. The proofs involve the application of a variety
of martingale techniques—most importantly a “spine” construction
involving a change of measure with an additive martingale. In ad-
dition to the model’s intrinsic interest, the methodologies presented
contain ideas that will adapt to other branching settings. We also
briefly discuss applications to traveling wave solutions of an associ-
ated reaction–diffusion equation.
1. Introduction. In this article we will consider a certain family of typed
branching diffusions that have particles which move (independently of each
other) in space according to a Brownian motion with variance controlled by
the particle’s type process. The type of each particle evolves as an Ornstein–
Uhlenbeck process and this type also controls the rate at which births occur.
The particular form of this model permits many explicit calculations, but
throughout we will strive to develop techniques that rely on general prin-
ciples as much as possible, so they might readily adapt to other situations.
This model was previously considered in [12, 13]; these papers form essential
foundations for this work, although we will recall various results as necessary.
Received December 2004; revised November 2006.
1Supported in part by an EPSRC studentship.
AMS 2000 subject classification. 60J80.
Key words and phrases. Spatial branching process, branching diffusion, multi-type
branching process, additive martingales, spine decomposition.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2007, Vol. 17, No. 2, 609–653. This reprint differs from the original in pagination
and typographic detail.
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2 Y. GIT, J. W. HARRIS AND S. C. HARRIS
We will make some significant applications of the spine theory for branch-
ing processes. Inspired by the series of papers Lyons, Pemantle and Peres
[19], Lyons [18] and Kurtz, Lyons, Pemantle and Peres [16], spine techniques
have been instrumental in recent years in providing intuitive and elegant
proofs of many important classical and new results in the theory of branch-
ing processes. In this article we use the recent reformulation of the spine
method presented in [8], which follows in similar spirit to the branching
Brownian motion study of Kyprianou [17]. For a selection of other applica-
tions of spine techniques, for example, see [1, 6, 7, 23] and references therein.
1.1. The branching model. We define Nt to be the set of particles alive
at time t ≥ 0. For a particle u ∈ Nt, Xu(t) ∈ R is its spatial position, and
Yu(t) ∈ R is the type of u. We will label offspring using the Ulam–Harris
convention where, for example, if u=∅21 then particle u is the first child
of the second child of the initial ancestor, and we will write v > u if particle
v is a descendant of particle u. The configuration of the branching diffusion
at time t is given by the point process Xt := {(Xu(t), Yu(t)) :u ∈Nt}.
A particle’s type evolves as an Ornstein–Uhlenbeck process with an invari-
ant measure given by the standard normal density φ(y) and an associated
differential operator (generator)
Qθ :=
− y ∂
where θ > 0 is considered to be the temperature of the system. The spatial
motion of a particle of type y is a driftless Brownian motion on R with
variance
A(y) := ay2, where a≥ 0.
A particle of type y particle is replaced by two offspring at a rate
R(y) := ry2 + ρ, where r, ρ≥ 0.
Each offspring inherits its parent’s current type and spatial position, and
then moves off independently of all others. We use P x,y and Ex,y with x, y ∈
R to represent probability and expectation when the Markov process starts
with a single particle at position (x, y).
We will find the almost sure rate of exponential growth, D(γ,κ), of par-
ticles which are found simultaneously with spatial positions near −γt and
type positions near κ
t at large times t. From this we can deduce the speed
of extremal particles and hence the asymptotic shape of the particle system.
The main effort is required in identifying D(γ,κ) as the almost sure limit of
t−1 log
1{Xu(t)≤−γt;Yu(t)≥κ
A TYPED BRANCHING DIFFUSION 3
In particular, the convergence properties of two different families of addi-
tive martingales associated with the branching diffusion will lead directly to
the spatial exponential growth rates and an upper bound on the space-type
growth. For the remaining lower bound, we describe an explicit two-phase
mechanism for amassing the required number of particles with prescribed
space-type positions. The first phase involves building up an “excess” num-
ber of particles, each covering a certain proportion of the required spatial
distance. During their second phase, enough of these particles must succeed
in making a difficult and rapid ascent into the required position. The latter
phase is proved using an intuitive change of measure technique that induces
a spine construction.
The family of models we are considering is specific but nevertheless have
some features of fundamental significance that motivate the choices for Qθ,
R and A. If the spatial motion is ignored, we have investigated a binary
branching Ornstein–Uhlenbeck process in a quadratic breeding potential. In
contrast, Enderle and Hering [5] considered a branching Ornstein–Uhlenbeck
with constant branching rate but random offspring distribution. A quadratic
breeding potential is a critical rate for explosions in the population of parti-
cles. In a branching Brownian motion on R with binary splitting occurring
at rate xp at position x, the population will explode almost surely in finite
time if p > 2, whereas for p = 2 the expected number of particles explodes
while the total population remains finite for all time with probability 1 (see
[15], Chapter 5.12). The Ornstein–Uhlenbeck process is not only a canoni-
cal ergodic diffusion, but this type-motion has exactly the right drift to help
counteract the quadratic breeding rate. For high temperatures, θ > 8r, there
is a sufficiently strong mean-reversion in the type processes to ensure that
the expected total population size does not blow up; but for temperatures
θ ≤ 8r, the quadratic breeding overpowers the pull toward the origin, the
expected population blows up in a finite time and particles behave very dif-
ferently. Throughout this paper we consider only high temperatures θ > 8r,
deferring the low and critical temperature regimes to future work. Given
other choices, the quadratic spatial diffusion coefficient now becomes very
natural, enabling us to find explicit families of (fundamental) additive mar-
tingales since the linearized traveling-wave equation can be linked to the
classical harmonic oscillator equations from physics. The binary branching
mechanism was taken for simplicity; in principle our approach could extend
to general offspring distributions, although new features would arise from
possible extinctions and necessary offspring moment conditions. All these
choices make the models rich in structure, possessing some very challenging
features whilst remaining sufficiently tractable.
4 Y. GIT, J. W. HARRIS AND S. C. HARRIS
1.2. Application to reaction–diffusion equations. Following in the foot-
steps of McKean [20], the solution of the reaction–diffusion equation
+R(y)u(u− 1) + θ
− y∂u
with initial condition f(x, y)∈ [0,1] for all x, y ∈R, can be represented by
u(t, x, y) =Ex,y
f(Xu(t), Yu(t))
Of great importance for reaction–diffusion equations are traveling-wave so-
lutions (e.g., see [21]). In the present context, a solution to equation (1) of
the form u(t, x, y) := w(x− ct, y) is said to be a traveling-wave of speed c,
where w(x, y) solves the traveling-wave equation
+R(y)w(w − 1) + θ
− y∂w
= 0.(3)
Fundamental to our study of the branching diffusion are two families of “ad-
ditive” martingales, Z±λ (t) [defined at (6)], which are linked to the lineariza-
tion of (1). When θ > 8r, Harris and Williams [13] determined when Z−λ is
uniformly integrable (see Theorem 17) and then wλ(x, y) :=E
x,y exp(−Z−λ (∞))
yields a traveling wave of speed c−λ . This gives the existence of traveling
waves for all speeds c greater than some threshold c̃(θ) := inf c−λ .
Furthermore, combining the McKean representation (2) with the almost-
sure convergence result established in [12] (look ahead to Theorem 18) can
give results on the attraction toward traveling waves from given initial data.
For example, if − lnf(x, y)∼ eλxg(y) uniformly in y as x→∞ for some suit-
able g ∈ L2(φ), the solution u(x, y) to (1) with initial conditions f satisfies
u(t, x− c−λ t, y)→wλ(x+ x̂, y) as t→∞, where x̂ is some constant that can
be determined from g.
In future work we hope to develop the approach used for standard BBM
and the FKPP equation in [11], and prove that traveling waves of a given
speed c > c̃(θ) are unique (up to translation) and that no traveling waves
exist for speeds c < c(θ). We anticipate that our new results on the growth
rates of particles will aid in establishing some difficult estimates on the tail
behavior of any traveling wave, and hence assist in proving the conjectured
uniqueness. In addition, we expect our growth rate results will be essential
in obtaining broader classes of initial conditions that are attracted toward
traveling waves. In each of these problems, difficulties arise from the un-
bounded type space where, for example, some control must be gained over
the possible contributions to
u∈Nt log f(Xu(t) − ct, Yu(t)) from particles
that have large type positions in addition to large spatial positions.
A TYPED BRANCHING DIFFUSION 5
2. Main results. In this section, we will present our main results that
identify the growth rates found within the branching diffusion. We will give
an overview for our proofs, identifying the key ideas and techniques used,
as well as introducing some intuition for the dominant behavior of particles
that underpins our approach.
2.1. Martingales. The principal tools used throughout this paper are
two fundamental families of “additive” martingales, which were introduced
in [13].
Before defining the martingales we give some key definitions. Let
λmin :=−
θ− 8r
Let λ ∈R, with the following convention which we always use for λ:
λmin <λ< 0.
Also, define
µλ :=
θ(θ− 8r− 4aλ2), ψ±λ :=
E±λ := ρ+ θψ
λ , c
λ :=−E
λ /λ.(5)
Will will occasionally write E±λ as E
±(λ) in order to emphasize that E±λ are
really functions of λ; the ± superscripts will always distinguish these from
expectation operators. Note that λmin is the point beyond which µλ is no
longer a real number.
The martingales are Z−λ and Z
λ , defined for λ ∈ (λmin,0] as
Z±λ (t) :=
v±λ (Yu(t))e
λXu(t)−E±λ t,(6)
where v±λ (y) := exp(ψ
2) are strictly-positive eigenfunctions of the opera-
Qθ + 12λ
2A+R,
with corresponding eigenvalues E−λ <E
λ and A,R are the functions defined
in Section 1.1. This operator is self-adjoint on L2(φ) with the inner product
〈·, ·〉φ where 〈f, g〉φ :=
fgφdy and φ is the standard normal density. Note
that v−λ ∈ L2(φ), whereas v
λ /∈L2(φ) so is not normalizable.
The calculations of Section 3 make it easy to see these are martingales,
and throughout the paper we will need a variety of martingale convergence
results which are gathered together in Section 8. In particular, we will need
to know precisely when Z−λ is uniformly integrable with a strictly positive
limit, some further strong convergence results for other closely related sums
over particles (also identifying which particles contribute nontrivially to their
limits), and the rate of convergence to zero of the Z+λ martingales.
6 Y. GIT, J. W. HARRIS AND S. C. HARRIS
2.2. The asymptotic growth-rate of particles along spatial rays. As an
essential initial step toward determining the growth rate of particles in the
two-dimensional space-type domain, we first look at the growth rate of par-
ticles in the spatial dimension only.
For γ ≥ 0 and C ⊂R, define
Nt(γ;C) :=
1{Xu(t)≤−γt;Yu(t)∈C}.(7)
The limit giving the expected rate of growth,
t−1 logE(Nt(γ;R))
can be shown to exist and its value can be calculated to be
∆(γ) := inf
λ∈(λmin,0)
{E−λ + λγ}
a−1(θ − 8r)(4γ2 + θa).
An outline for this expectation calculation is given in Section 3.
It is now tempting to guess that the asymptotic speed of the spatially
left-most particle, c̃(θ), is given by
c̃(θ) := sup{γ :∆(γ)> 0}
r+ ρ+
2(2r+ ρ)2
θ− 8r
Recall that c̃(θ) = infλ∈(λmin,0) c
λ is also the minimum threshold for traveling
waves. In this particular situation, the guess that “expectation” and “almost
sure” right-most particle speeds agree was first proved rigorously using a
martingale change of measure technique in [13]. In this paper, we extend
this connection and prove that the “expected” and “almost sure” rates of
growth of particles with given speeds (Theorem 1) and given space-type
locations (Theorem 3) agree.
Theorem 1. Let γ ≥ 0 and y0 < y1. Under each P x,y law, the limit
D(γ) := lim
t−1 logNt(γ; [y0, y1])
exists almost surely and is given by
D(γ) =
∆(γ), if 0≤ γ < c̃(θ),
−∞, if γ ≥ c̃(θ).
A TYPED BRANCHING DIFFUSION 7
Note that symmetry in the process means there is a corresponding re-
sult for particles with spatial velocities greater than +γ (corresponding to
positive λ values). We may occasionally make use of such process symme-
tries without further comment. Then, since Nt(γ;R) is integer valued, the
asymptotic speed of the right-most particle follows immediately:
Corollary 2. Almost surely,
t−1 sup{Xu(t) :u ∈Nt}= c̃(θ).
This spatial growth rate result is proved in Section 10 using the martin-
gale results from Section 8. In fact, it is very easy to obtain the upper bound
by first dominating the indicator function with exponentials to reveal that
Nt(γ;R) ≤ exp{(E−λ + λγ)t}Z
λ (t), recalling that Z
λ is a convergent mar-
tingale, and then optimizing over the choice of λ. For the lower bound, we
will use a strong convergence result obtained in [12], combined with the idea
that each uniformly integrable martingale Z−λ essentially “counts” only the
particles of corresponding velocity −γ.
2.3. The asymptotic shape and growth of the branching diffusion. The
main result of this paper is the almost-sure rate of growth of particles which
are in the vicinity of −γt in space and near κ
t in type position at large
times t. For γ, κ≥ 0, it can be shown that the limit
t−1 logE(Nt(γ; [κ
t,∞)))(10)
exists and takes the value
∆(γ,κ) := inf
λ∈(λmin,0)
{E−λ + λγ − κ
2ψ+λ }
(θ− κ2)
θ(θ− 8r)(4aθγ2 + a2(θ+ κ2)2).
An outline of this expectation calculation is given in Section 3. Once again,
we will find that the “almost sure” rate of growth of particles agrees with
this “expected” rate exactly where there is growth in particle numbers.
Theorem 3. Let γ,κ ≥ 0 with ∆(γ,κ) 6= 0. Under each P x,y law, the
limit
D(γ,κ) := lim
t−1 logNt(γ; [κ
t,∞))
exists almost surely and is given by
D(γ,κ) =
∆(γ,κ), if ∆(γ,κ)≥ 0,
−∞, if ∆(γ,κ)< 0.(12)
8 Y. GIT, J. W. HARRIS AND S. C. HARRIS
To prove the tricky lower bound of Theorem 3, which amounts to the
major work of this paper, we will exhibit an explicit two-phase mechanism by
which the branching diffusion can build up at least the required exponential
number of particles near to −γt in space and κ
t in type position by large
times t.
During the first phase, over a large time t the process builds up an ini-
tial excess of approximately exp(∆(α)t) particles with spatial position at
least −αt, as is already known from Theorem 1. In this “ergodic” phase,
“typical” particles found near −αt in space will have drifted with a steady
spatial speed of α whilst their type histories will have behaved roughly like
OU processes with inward drift of µλy for a certain optimal choice λ(α) of
parameter λ.
For the second phase, we will show that the probability any individual
particle has at least one descendant that makes a “rapid ascent” in both
space and type dimensions from initial position (0,0) to final position near
(−βt,κ
t) is approximately exp(−Θ(β,κ)t), where
Θ(β,κ) =
θ(θ− 8r)(a2κ4 +4aθβ2)
,(13)
and the time taken for this “rapid ascent” is an interval [0, τ ]. We show that
this time τ can be chosen such that 2µλτ ∼ log t, and hence the additional
time is asymptotically negligible in comparison with t. Intuitively, we will
see that given an offspring that has successfully made such a difficult “rapid
ascent,” it will most likely have roughly had its type process behaving like an
OU process with an outward drift of µλy and the Brownian motion driving
its spatial motion will have had a drift λ [corresponding to a real time spatial
drift λA(y) that increases in strength as the type position y increases], for
some optimal choice λ(β,κ) of parameter λ. The precise result required will
be formulated rigorously as a large-deviation lower bound in Theorem 7
of Section 5, and is proved using a “spine” change of measure technique
intimately related to the Z+λ martingales.
Combining these two phases and using independence of the particles, we
can see that the number of particles near (−αt,0) at time t that subse-
quently proceed to have at least one descendant near (−(α + β)t, κ
t) is
approximately Poisson with mean
exp({∆(α)−Θ(β,κ)}t).
Optimizing for a fixed overall spatial speed γ, some calculus reveals that
α+β=γ
α,β>0
{∆(α)−Θ(β,κ)}=∆(ᾱ)−Θ(β̄, κ) = ∆(γ,κ),(14)
A TYPED BRANCHING DIFFUSION 9
with optimal parameters
ᾱ= γ
θ+ κ2
and β̄ = γ
θ+ κ2
.(15)
Thus we will be able to demonstrate an explicit two-phase mechanism pro-
ducing the required number of particles, with this outline argument later
guiding our rigorous proof. In addition, it is interesting to note that the
optimal choices for λ over each phase then also coincide at a single value
λ̄= λ(ᾱ) = λ(β̄, κ).
An informative large deviation heuristic for the rapid ascent can also be
found in Section 4, with this section also containing some essential optimal
path calculations. We actually prove the two-phase mechanism for the lower
bound of Theorem 3 in Section 5, although we defer proving the large-
deviation lower bound until Section 7 after presenting the necessary “spine”
background in Section 6.
We prove the upper bound for the space-type growth rate in Section 9,
again making crucial use of martingale results from Section 8. Similarly to
the spatial growth case, we can find an upper bound using the Z+λ mar-
tingales, that is to say Nt(γ; [κ
t,∞)) ≤ exp{(E+λ + λγ − κ2ψ
λ )t}Z
λ (t).
However, as each Z+λ martingale converges to zero, we must show that its
exponential decay rate is (E+λ −E
λ ) before being able to optimize over the
choice of λ to obtain the required upper bound.
Given Theorem 3, and noting symmetries, it becomes straightforward to
retrieve the following:
Corollary 4. For any F ⊂R2, define
Nt(F ) :=
1{(Xu(t)/t,Yu(t)/
t)∈F}.
If B ⊂R2 is any open set and C ⊂R2 is any closed set, then almost surely
under any P x,y
lim inf
logNt(B)≥ sup
(γ,κ)∈B
D(γ,κ),
lim sup
logNt(C)≤ sup
(γ,κ)∈C
D(γ,κ),
with the growth rate D(γ,κ) given at equation (12).
We can also recover the almost sure asymptotic shape of the region occu-
pied by the particles in the branching diffusion.
10 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Corollary 5. Let B ⊂R2 be any open set. Almost surely, under each
P x,y law,
Nt(B)→
0, if S ∩B =∅,
+∞, if S ∩B 6=∅,
where S ⊂R2 is the set given by
S := {(γ,κ) ∈R2|∆(γ,κ)> 0}.
3. Some expectation calculations. This section discusses how the ex-
pected growth rates given in the previous section may be obtained. For this,
we use the “many-to-one” lemma (see, e.g., [8]) and one-particle changes of
measure. In the process we shall start to gain valuable intuition into how
particles within the branching diffusion behave, as well as seeing hints as to
which are the “correct” martingales to use to prove the almost-sure growth
rate results.
For simplicity, we assume throughout this section that the branching dif-
fusion starts with one particle at the origin in both space and type at time
zero, unless otherwise stated. We also introduce a family of single particle
probability measures Pµ,λ with associated expectations Eµ,λ where, under
Pµ,λ, η is an Ornstein–Uhlenbeck process with variance θ and drift µ, and
ξt =B(
0 A(ηs)ds) where B is a Brownian motion with drift λ.
Lemma 6 (Many-to-one). If f :R2 7→R is Borel measurable then
f(Xu(t), Yu(t)) = Eθ/2,0
R(ηs)ds
f(ξt, ηt)
.(16)
Using the many-to-one lemma, and changing measure to alter the drift of
Brownian motion, we see that
f(Xu(t), Yu(t))
= Eθ/2,0
R(ηs)ds
f(ξt, ηt)
= Eθ/2,0
e−λξt exp
R(ηs) +
A(ηs)
× f(ξt, ηt) · eλξt−λ
A(ηs)ds
= Eθ/2,λ
−λξt +
R(ηs) +
A(ηs)
f(ξt, ηt)
A TYPED BRANCHING DIFFUSION 11
To perform a further change of measure on the OU process to get rid of the
time integrals in the exponential of the expectation, we recall that
dPµλ,·
dPθ/2,·
µλ,θ/2
:= exp
ψ−λ η
t −E−λ t+
R(ηs) +
λ2A(ηs)
and then
f(Xu(t), Yu(t))
= Eθ/2,λ(exp(−λξt −ψ−λ η
λ t)f(ξt, ηt) ·M
µλ,θ/2
t )(17)
= Eµλ,λ(exp(−λξt − ψ
λ t)f(ξt, ηt)).
Note that the many-to-one lemma, combined with the branching prop-
erty, immediately suggests how to get “additive” martingales for the branch-
ing diffusion from single particle martingales—for example, taking f(x, y) =
exp{λx+ ψ−λ y2} in equation (17) quickly leads to the martingale Z
We may now proceed to calculate the expected growth rates. However,
for both clarity and brevity we will leave rigorous details to the interested
reader, noting that the intuition we will gain from our rough calculations
will later be invaluable in guiding our rigorous proof of the corresponding
almost-sure growth rates.
3.1. The expected rate of growth along spatial rays. We first give the
outline of some calculations to find the rate of growth in the expected number
of particles near −γt in space at time t.
Using the formula from (17), for λ ∈ (λmin,0) and any ε > 0 we have
1{t−1Xu(t)+γ∈(−ε,ε)}
= Eµλ,λ(e
−λξt−ψ−λ η
1{t−1ξt+γ∈(−ε,ε)})
≤ e(E
+λγ−λε)tEµλ,λ
+ γ ∈ (−ε, ε)
≥ e(E
+λγ+λε)tEµλ,λ
η2t ;
+ γ ∈ (−ε, ε)
where, with some abuse of notation that we shall continue to use throughout
this section, we will abbreviate this to
1{Xu(t)∼−γt} = Eµλ,λ(e
−λξt−ψ−λ η
1{ξt∼−γt})
∼ e(E
+λγ)t
Eµλ,λ(e
t ; ξt ∼−γt)
12 Y. GIT, J. W. HARRIS AND S. C. HARRIS
with the understanding that any subsequent arguments to identify expo-
nential growth rates can readily be made rigorous by using the appropriate
upper and lower bounds, and so on.
Now, considering E−(λ) :=E−λ as a function of λ, we have from (8) that
∆(γ) = infλ∈(λmin,0){E−(λ) + λγ}=E−(λγ) + λγγ, where λγ satisfies
(λγ) =−γ, hence λγ =−γ
(θ− 8r)
θa2 +4aγ2
.(19)
Of course, choosing this optimal λγ value in (18) means that we must have si-
multaneously maximized the expectation Eµλ,λ(exp(−ψ
t ); ξt ∼−γt), and
to confirm that this value is not exponentially decaying in t is now relatively
straightforward. Under Pµλ,λ, η is an Ornstein–Uhlenbeck process with an
invariant measure given by the probability density, φλ, of the normal distri-
bution N(0, θ/(2µλ)); and ξt =B(
0 A(ηs)ds), where B is a BM with drift
λ. Note also that by differentiating
〈(Qθ + (1/2)λ2A+R−E−λ )v
λ , v
λ 〉φ = 0
with respect to λ, using self-adjointness, and observing that φλ ∝ (v−λ )2φ,
we find that
〈λAv−λ , v
〈v−λ , v
A(y)φλ(y)dy.
Then almost surely under Pµλ,λ,
0 A(ηs)ds)
0 A(ηs)ds
0 A(ηs)ds
Aφλ dy =
,(20)
and so when we use the optimal λγ value we get exactly the desired drift,
since ∂E
(λγ) =−γ. Then
Eµλγ ,λγ (e
η2t ; ξt ∼−γt) → lim
Eµλγ ,λγ(e
η2t )
φλγ (y)dy.
In this way, we can obtain the exact rate of exponential growth for the
expectation,
t−1 logE(Nt(γ;R)) = ∆(γ).
The changes of measure used above are actually suggesting a great deal
about the dominant particles that are found in the vicinity of a given ray
in space. An alternative discussion of this expectation result, involving a
dual approach via large deviation theory for occupation densities, can also
be found in [13].
A TYPED BRANCHING DIFFUSION 13
3.2. The expected asymptotic shape. We give a rough outline of calcula-
tions that will yield the correct exponential growth in the expected number
of particles both near −γt in space and κ
t in type at large times t. Using
the formula from (17) and abusing notation throughout in the same way as
Section 3.1, we find that
1{Xu(t)∼−γt;Yu(t)≥κ
= Eµλ,λ(e
−λξt−ψ−λ η
1{ξt∼−γt;ηt≥κ
∼ e(E
+λγ−κ2ψ−
Pµλ,λ(ξt ∼−γt;ηt ≥ κ
Now, from standard bounds on the tail of the normal distribution,
Pµλ,λ(ξt ∼−γt;ηt ≥ κ
= Pµλ,λ(ηt ≥ κ
t)Pµλ,λ(ξt ∼−γt|ηt ≥ κ
t)(21)
∼ e−µλ/θκ2tPµλ,λ(ξt ∼−γt|ηt ≥ κ
and, since ψ−λ + (µλ/θ) = ψ
λ , this yields
1{Xu(t)∼−γt;Yu(t)≥κ
∼ e(E
+λγ−κ2ψ+
Pµλ,λ(ξt ∼−γt|ηt ≥ κ
Recalling that ∆(γ,κ) := infλ∈(λmin,0){E
λ +λγ−κ2ψ
λ }, simple calculus re-
veals this infimum is attained at a λ value of
λ̄(γ,κ) =−γ
θ(θ− 8r)
a2(κ2 + θ)2 + 4aγ2θ
∈ (λmin,0),(23)
and using this optimal value in equation (22) will lead to the upper bound
limsup
t−1 logE
1{Xu(t)∼−γt;Yu(t)≥κ
t} ≤∆(γ,κ).
It is also clear from equation (22) that when minimizing E−λ + λγ − κ2ψ
we simultaneously maximize the probability Pµλ,λ(ξt ∼ −γt |ηt ≥ κ
t). In
particular, to get a matching lower bound, we do not want this probability
to have any exponential decay in time when we choose the optimal parameter
for λ.
In fact, at least up to the exponential decay rate in time, it can be shown
using large-deviations arguments that
Pµλ̄,λ̄
(ξt ∼−γt;ηt ≥ κ
t)∼ exp
14 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Indeed, we immediately gain the required upper bound from (21). For the
lower bound, consider the following heuristics where we break paths into two
sections: normal ergodic behavior over large time period [0, t] followed by a
rapid ascent out to type position κ
t over a much shorter period [t, t+ τ ].
(i) Ergodic behavior. Over a large time t, the occupation density of η
will most likely have settled close to the invariant measure. Hence for large
t, almost surely under Pλ,µλ ,
η2s ds→
(ii) Rapid ascent. Over a large time τ , but where τ = o(t), the probability
that η starts close to the origin and ends near to κ
t, having followed close
to the path y over the entire time period τ , is roughly given by
{ẏ(s) + µλy(s)}2 ds
under the Pλ,µλ law. See, for example [24], Chapter 6, or [4], Chapter 5.6.
After some Euler–Lagrange optimization, the path
y(s) = κ
sinhµλs
sinhµλτ
gives
0 y(s)
2 ds≈ κ2t/(2µλ), with the probability of this path being roughly
exp(−(µλ/θ)κ2t).
Combining these two types of behavior, we can find paths with final positions
ηt+τ ∼ κ
t, ξt+τ ∼ λa
∫ t+τ
η2s ds∼ λa
and, moreover, when substituting the optimal λ value of λ̄(γ,κ) and sim-
plifying, this actually gives ξt+τ ∼ −γt. Further, one of these paths occurs
with a probability of roughly exp(−(µλ̄/θ)κ2t), and note that t+ τ ∼ t since
τ = o(t). Thus we see that to exponential order, the probability Pµλ̄,λ̄
(ξt ∼
−γt;ηt ≥ κ
t) must be at least exp(−(µλ̄/θ)κ2t), as required.
This heuristic argument can be made rigorous to prove, as claimed, that
t−1 logE
1{Xu(t)≤−γt;Yu(t)≥κ
=∆(γ,κ).
If we scale all spatial coordinates by t−1 and all type coordinates by
t)−1 at time t, the expected asymptotic shape can be considered to be the
region S := {(γ,κ) :∆(γ,κ) ≥ 0} where, on average, we have growth in the
numbers of (scaled) particles.
A TYPED BRANCHING DIFFUSION 15
4. Short climb large deviation heuristics. In this section, we give a heuris-
tic calculation that suggests why the probability a single particle manages
to have at least one descendant in the vicinity of (−βt,κ
t ) near time τ
is roughly exp(−Θ(β,κ)t) for very large t, where Θ(β,κ) is given at equa-
tion (13). For these heuristics, we will think of τ as large and fixed, but of
smaller order than t (later on, in our rigorous approach, we will choose τ
proportional to log t). We emphasize that the heuristics in this section are
neither meant to be precise nor made rigorous, yet they will provide invalu-
able intuition, guidance and motivation for our rigorous approach later on.
Of particular importance will be the optimization problem that the heuris-
tics suggest. Indeed, many of the exact calculations in Sections 4.2 and 4.3
will be essential later in the paper.
Suppose we start the branching diffusion with a single particle at (0,0).
First, we wish to know the probability that there is at least one particle at
time τ that has a spatial position near −βt having followed close to the path
x(s) for 0≤ s≤ τ and a type position near κ
t having closely followed the
path y(s) for 0≤ s≤ τ for t arbitrarily large.
We recall from large deviation theory of Ventcel–Freidlin (see [24], Chap-
ter 6, or [4], Chapter 5.6) that the probability a single particle manages to
follow closely both the type path y(s) and the spatial path x(s) for 0≤ s≤ τ
is roughly given by
ẏ(s) +
ds− 1
ẋ(s)2
ay(s)2
when x(0) = 0, x(τ) = −βt, y(0) = 0, y(τ) = κ
t and t is very large. This
probability will typically be very small, but if such paths are followed by
particles in the branching diffusion, we have to also take account of the
large breeding rates that are found far from the type origin.
If we let X(s) represent the numbers of particles in the branching diffusion
that are alive at time s and have traveled “close” to the path (x(u), y(u))
for 0 ≤ u ≤ s, then we can get a rough idea of how X might behave by
considering the following birth–death process.
4.1. A birth–death process. For given fixed paths x(·) and y(·), let M be
a time-dependent birth–death process where at time s particles either give
birth to single offspring with breeding rate λ(s) given by
λ(s) = ρ+ ry(s)2,
or particles die with death rate µ(s) given by
µ(s) =
ẏ(s) +
ẋ(s)2
ay(s)2
16 Y. GIT, J. W. HARRIS AND S. C. HARRIS
(Note that the probability the initial particle of this birth–death process sur-
vives the entire time period [0, τ ] is consistent with the rough large deviation
probability for the branching diffusion at equation (24).)
An important quantity is the effective total death rate up to time t which
is defined by ν(s) :=
0 {µ(w)− λ(w)}dw, so here
ν(s) = J(x, y, s)
ẏ(w) +
ẋ(w)2
ay(w)2
− ry(w)2 − ρ
The distribution for total number of offspring surviving, M(τ), for the
time-dependent birth–death process is well known, for example, see [14].
Then defining
Wτ := e
−ν(τ)
µ(s) eν(s) ds
Uτ := 1− e−ν(τ)W−1τ ,
Vτ := 1−W−1τ ,
we have
P(M(τ) = 0) = Uτ ,
P(M(τ) = n) = (1−Uτ )(1− Vτ )V n−1τ , n= 1,2, . . .
with EM(τ) = e−ν(τ) and E(M(τ)|M(τ)≥ 1) =Wτ .
In our particular case, we have
E(M(τ)) = exp(−J(x, y, τ)).
Define the largest effective total death rate prior to time τ by
L(x, y, τ) := sup
s∈[0,τ ]
J(x, y, s)≥ 0.
If we are in a case when L(x, y, τ) is very large, suggesting a high chance of
extinction, then
P(M(τ)≥ 1) = 1
0 µ(s)e
ν(s) ds
∼Kτ exp(−L(x, y, τ)),(25)
where K−1τ :=
0 µ(s) exp(−{L(x, y, τ) − J(x, y, s)})ds. If there is at least
one particle alive, we would then expect to have
E(M(τ)|M(τ)≥ 1)∼K−1τ exp(L(x, y, τ)− J(x, y, τ)).
Thus, we might guess that the probability any particles in the branching
diffusion manage to make the difficult, rapid ascent along path (x, y) to finish
up near (−βt,κ
t ) can, very roughly, be estimated by exp(−L(x, y, τ)). [To
A TYPED BRANCHING DIFFUSION 17
help see this, try writing x(s) = tf(s) and y(s) =
tg(s), thinking of f, g as
fixed paths and recall that t is very large and τ = o(t), then the role of Kτ
in (25) is insignificant next to exp(−L(x, y, τ)).]
We might then further guess that the chance any particles manage to
stay near position (−βt,κ
t) during a very small interval of time close to τ
should roughly look like
− inf
L(x, y, τ)
where we permit all possible paths x and y satisfying x(0) = 0, x(τ) =−βt
and y(0) = 0, y(τ) = κ
t for the fixed time τ . (We will state and prove a
precise lower bound that corresponds to this guess at Theorem 7.)
4.2. Finding the optimal path and probability. We proceed to calculate
L(x, y, τ)
over paths x and y satisfying x(0) = 0, x(τ) =−βt and y(0) = 0, y(τ) = κ
for the fixed time τ .
We first note that
L(x, y, τ) = inf
s∈[0,τ ]
J(x, y, s)≥ inf
J(x, y, τ)(26)
and we now proceed to calculate infx,y J(x, y, τ).
We can easily optimize over the choice of function x given y, finding that
ẋ(s)∝ ay(s)2 ⇒ x(s) = λa
y(u)2 du
where λ is the constant of proportionality and must satisfy
0 y(s)
,(27)
yielding
ẋ(s)2
ay(s)2
0 y(s)
This is exactly as anticipated since, when following the path y in type space,
the spatial position of a particle is following a Brownian motion with total
amount of variance over period τ given by a
0 y(s)
2 ds. Hence, the proba-
bility that a particle following the path y in type space will also be found
near to βt in space at time τ is roughly
( −β2t2
0 y(s)
18 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Introducing the notation
I(y) :=
ẏ(s) +
− ry(s)2
we are left to find
I(y) +
0 y(s)
= inf
I(y)− 1
y(s)2 ds− λβt
≥ sup
I(y)− 1
y(s)2 ds− λβt
where the first equality is trivially true by maximizing the quadratic in
λ, the introduction of which conveniently removes the awkward integral in
the denominator. Some further Euler–Lagrange optimization now gives the
optimal path as
yλ(s) = κ
sinhµλs
sinhµλτ
(0≤ s≤ τ),(29)
where
θ(θ− 8r− 4aλ2)
and then
I(y)− 1
y(s)2 ds− λβt
= sup
cothµλτ
− λβt
The optimal parameter choice λ̂ (which depends on τ as well as the model
parameters) then satisfies
= κ2t
cothµ
2 sinh2 µ
(s)2 ds.(30)
Then we have shown that
0 yλ̂(s)
≥ inf
I(y) +
0 y(s)
= inf
I(y)− 1
y(s)2 ds− λβt
≥ sup
I(y)− 1
y(s)2 ds− λβt
≥ I(y
(s)2 ds− λ̂βt,
A TYPED BRANCHING DIFFUSION 19
and, in fact, we see that the left- and right-hand sides of the above are equal
by recalling (30). It follows that the preceding supremum and infimum can
be freely interchanged, actually preserving equality at the inequality (28).
Then, with the optimal spatial path
xλ(s) := λa
yλ(u)
2 du=−βt sinh2µλs− 2µλs
sinh2µλτ − 2µλτ
,(31)
and defining x̂ := x
, ŷ := y
, we have
J(x, y, τ) = J(x̂, ŷ, τ)
= t sup
cothµλτ
cothµ
− λ̂β
− ρτ.
Finally, it is easy to check that J(x̂, ŷ, τ) = L(x̂, ŷ, τ), whence
J(x, y, τ)≥ inf
L(x, y, τ),
and, combining with equation (26), we have found that
L(x, y, τ) = inf
J(x, y, τ) = J(x̂, ŷ, τ).
4.3. An important note on the optimal paths. As τ →∞, we have
cothµλτ
↑ sup
{κ2ψ+λ − λβ}= κ
− λ̄ β,
where the optimizing parameters of the supremums also converge with
λ̂→ λ̄=−β
θ(θ− 8r)
a2κ4 + 4aθβ2
κ2 + θ
.(32)
Note the agreement with previous optimal values at equations (23) and (15).
Then letting
Θ(β,κ) := sup
{κ2ψ+λ − λβ}
θ(θ− 8r)(a2κ4 +4aθβ2)
and writing x̄ := xλ̄ and ȳ := yλ̄, we note that for all ε, δ > 0 there exist
τ̃ , µ > 0 such that for all t > 0 and τ > τ̃
− inf
J(x, y, τ)
≥ exp(−J(x̄, ȳ, τ))
= exp
cothµλ̄τ
− λ̄β
≥ exp(−t(Θ(β,κ) + ε)).
20 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Further (when κ > 0), for all s ∈ [τ − µ, τ ],
ȳ(s)≥ (κ− δ)
t, x̄(s)≤−(β − δ)t.
In particular, the paths stay close to the required positions for some fixed
length of time with corresponding probability at least as large as required.
5. Proof of Theorem 3. Lower bound. In this section we will state a pre-
cise short climb probability result and show how to combine it with almost
sure spatial (only) growth rates to prove the lower bound of the growth rate
in Theorem 3. This will make rigorous the two-phase mechanism described
in Section 2 and suggested by the expectation calculations in Section 3.
The first phase requires knowledge of the almost-sure rates of growth of
particles in the spatial dimension only. To this end, we will already make full
use of Theorem 1 throughout this section, deferring its proof until Section 10.
The second phase requires a lower bound for the probability that a single
particle makes a rapid ascent in type-space over the time interval [0, τ ]. This
is the lower bound found in the heuristics of Section 4, but we require some
further notation before the precise result can be stated. Note, throughout
this section, we will only be interested in the optimal parameter value λ= λ̄
as introduced in Section 4.3.
We wish to fix the relationship between sufficiently large t and τ as
θ/(2µλ̄) e
µλ̄τ = κ
t(34)
and so define τ = τ(t) by
τ(t) :=
(2µλ̄)
−1 log(2µλ̄t/θ), for 2µλ̄t > θ,
0, otherwise.
Recall the optimal paths (x̄, ȳ) over s ∈ [0, τ ], where
ȳ(s) = κ
sinhµλ̄s
sinhµλ̄τ
,(36)
x̄(s) = aλ̄
ȳ(w)2 dw=−βt sinh2µλ̄s− 2µλ̄s
sinh2µλ̄τ − 2µλ̄τ
,(37)
with fixed end points ȳ(τ) = κ
t and x̄(τ) =−βt.
For large times t and δ, ε > 0, let
t (u) :=
s∈[0,τ(t)]
|Yu(s)− ȳ(s)|< ε
t; sup
s∈[0,τ(t)]
|Xu(s)− x̄(s)|< δt
.(38)
We will use the notation
u∈Nτ(t)
t (u)(39)
A TYPED BRANCHING DIFFUSION 21
for the event that there exists a particle in the branching diffusion that
makes the short climb. Finally, recalling Θ(β,κ) given at (33), we can now
state the short climb theorem:
Theorem 7. Fix any y1 > y0 > 0, x ∈ R, and let ε0 > 0. Then for any
ε, δ > 0, there exists T > 0 such that for all y ∈ [y0, y1],
t−1 logP x,y(Aε,δt )≥−(Θ(β,κ) + ε0)
for all t > T .
We will prove Theorem 7 using a spine change of measure. This requires
us to introduce the notation for the spine set-up in detail before proceeding,
so this and further technical issues are postponed to Sections 6 and 7.
Remark 8. We note that Theorem 7 is actually a stronger result than
needed to prove Theorem 3 because we identify the specific paths followed
by particles that are near position (βt,κ
t ) at time t+ τ , rather than just
considering the particle’s positions close to time t+ τ .
In combining the two phases, we will have a huge number of independent
trials each with a small probability of success, intuitively giving rise to a
Poisson approximation for a large number of successful particles. In fact,
in our proof of the lower bound of Theorem 3 below, we will actually use
the following result about the behavior of sequences of sums of independent
Bernoulli random variables.
Lemma 9. For each n, define the random variable Bn :=
u∈Fn 1En(u)
where the events {En(u) :u ∈ Fn} are independent. Let pn(u) := P (En(u))
and Sn :=
u∈Fn pn(u) and suppose that, for some ν ∈ (1/2,1),
(Sn)2ν−1
<∞.(40)
Then the sequence of (possibly dependent) random variables {B1,B2, . . .}
has |Bn − Sn|> (Sn)ν for only finitely many n, almost surely.
In particular, for any ε > 0, there exists some (random) N ∈N such that,
with probability one,
> 1− ε for all n >N.(41)
Proof. For ν ∈ (1/2,1), Chebyshev’s inequality yields
P(|Bn − Sn|> (Sn)ν)≤
u∈Fn pn(u)(1− pn(u))
2ν−1 ,
22 Y. GIT, J. W. HARRIS AND S. C. HARRIS
and hence the Borel–Cantelli lemmas, combined with hypothesis (40), imply
|Bn − Sn|> (Sn)ν
for only finitely many n, almost surely. Equation (41) now follows on division
by Sn, and noticing the assumption (40) implies that limn→∞Sn =∞. �
Proof of Theorem 3. Lower bound. Define f−1(t) := t− τ(t), not-
ing that both f(t)/t→ 1 and f−1(t)/t→ 1 as t→∞. Also, for n ∈ N and
µ > 0, define Tn := (n+ 1)µ. We want to estimate the number of particles
that are near the large position (−(α + β)Tn, κ
Tn) during time interval
[Tn−1, Tn]. For this, we will consider particles that travel with a velocity
−α over time period [0, f−1(Tn)] before commencing their rapid ascent of
(relatively short) duration τ(Tn) to be in final position at time Tn. Then
s∈[Tn−1,Tn]
Ns((α+ β − δ)Tn; [(κ− δ)
Tn,∞))
u∈NTn
s∈[Tn−1,Tn]
{Xu(s)≤−(α+β−δ)Tn ;Yu(s)≥(κ−δ)
Tn}}(42)
u∈Fαn
1{N̄β,κn (u)>0}
where
Fαn := {u ∈Nf−1(Tn) :Xu(f
−1(Tn))≤−αTn, Yu(f−1(Tn)) ∈ [y0, y1]}
and, for u ∈ Fαn ,
N̄β,κn (u) :=
v∈NTn
s∈[Tn−1,Tn]
{Xv(s)−Xv(f−1(Tn))≤−(β−δ)Tn ;Yv(s)≥(κ−δ)
Tn}}.
We will now show that the sum at (42) grows as fast as anticipated:
Lemma 10. For any ε > 0, we may choose µ > 0 such that there exists
a random N ∈N where
u∈Fαn
1{N̄β,κn (u)>0}
≥∆(α)−Θ(β,κ)− ε
for all n >N with probability one.
Proof. We will be able to apply Lemma 9 given sufficient information
about the growth of |Fαn | and decay of the probabilities
pβ,κn (u) := P (N̄
n (u)> 0|Ff−1(Tn)),
A TYPED BRANCHING DIFFUSION 23
where u ∈ Fαn ⊂Nf−1(Tn).
It follows easily from Theorem 1, f−1(Tn)/Tn → 1 and the continuity of
∆(α) that
log |Fαn |
≥∆(α)− ε
for all sufficiently large n.
The definition of N̄β,κn (u) and spatial translation invariance implies that,
for each u ∈ Fαn , the rapid ascent probability pβ,κn (u) depends only on the
initial type position Yu(f
−1(Tn)).
For δ,µ > 0, define
t (u) :=
s∈[τ(t)−µ,τ(t)]
{Xu(s)−Xu(0)<−(β − δ)t;Yu(s)≥ (κ− δ)
u∈Nτ(t)
t (u).(43)
Recalling the comments of Section 4.3, there exist ε′, δ′ > 0 and we may
choose µ> 0 sufficiently small, such that
pβ,κn (u) = P
0,Yu(f
−1(Tn))(B
)≥ P 0,Yu(f−1(Tn))(Aε
) =: p̄n(u)
for all u ∈ Fαn whenever n is sufficiently large. Together with Theorem 7 and
since Yu(f
−1(Tn)) ∈ [y0, y1] for u ∈ Fαn , this reveals
log pβ,κn (u)
≥ log p̄n(u)
≥−Θ(β,κ)− ε
for all for u ∈ Fαn and all sufficiently large n, almost surely. Then we may
combine the observations above to obtain
u∈Fαn
pβ,κn (u)≥∆(α)−Θ(β,κ)−
Taking this last line together the assertion of Lemma 9 at equation (41)
gives the result. �
It is now straightforward to combine Lemma 10 with the inequality at
(42) to see that, given ε, δ > 0, there exists µ > 0 and a random time T such
t−1logNt((α+ β − δ)t; [(κ− δ)
t,∞))≥∆(α)−Θ(β,κ)− ε
for all t > T , almost surely. Since ε and δ can be taken arbitrarily small,
using the optimal ᾱ and β̄ according to equations (14)–(15), we find
lim inf
t−1 logNt(γ, [κ
t,∞))≥∆(γ,κ) almost surely,
24 Y. GIT, J. W. HARRIS AND S. C. HARRIS
as required. [It is also interesting to note that λ̄= λᾱ = λ̄(γ,κ) from equa-
tions (19), (23) and (32), so the optimal parameters are in agreement with
those of the expectation calculations in Section 3 and the path large devia-
tions in Section 4.] �
6. The “spine” setup and results. In this section, we describe how to
construct an enriched branching diffusion with an identified “spine” or “back-
bone” particle and discuss how to perform some extremely useful changes
of measure (closely related to the additive martingales) that will essentially
“force” the spine perform the short climb, whilst giving birth at an accel-
erated rate to offspring that behave as if under the original measure. These
spine techniques are at the very heart of our proof of Theorem 7 in Sec-
tion 5. Spine ideas were first seen for branching Brownian motion in [3] and
developed for Galton–Watson processes in [16, 18, 19]. Kyprianou [17] and
Englander and Kyprianou [6], developed the technique for some families of
branching diffusions; and more recently the spine approach has been signif-
icantly improved in [8]. This approach uses several different filtrations on
an enlarged probability space carrying the branching diffusion, and permits
some very useful techniques and results to be developed. For example, “addi-
tive” (many-particle) martingales can be represented as suitable conditional
expectations of “spine” (single-particle) martingales and consequently there
are clear interpretations for any changes of measure and all measures in-
volved in our “spine” setup are probability measures with intuitive construc-
tions. Following Hardy and Harris [8], we will first outline the notation and
then describe the changes of measure. The notation described in this section
is generalized to allow each particle u to have 1 +Au offspring, where each
Au is an independent copy of a random variable with values in {0,1,2, . . .}.
The spine techniques developed in this paper could readily be generalized
to such models.
All probability measures are to be defined on the space T̃ of marked
Galton–Watson trees with spines; before defining precisely what this space
is we need to set up some other notation. We recall the set of Ulam–Harris
labels, Ω, defined by Ω := {∅}∪
n∈N(N)
n, where N := {1,2,3, . . .}. For two
words u, v ∈Ω, uv denotes the concatenated word, where we take u∅=∅u=
u. So Ω contains elements such as “∅412,” which represents “the individual
being the 2nd child of the 1st child of the 4th child of the initial ancestor
∅.” For labels u, v ∈Ω the notation v < u means that v is an ancestor of u,
and |u| denotes the length of u.
We define a Galton–Watson tree to be a set τ ⊂Ω such that:
(i) ∅ ∈ τ , so there is the unique initial ancestor;
(ii) if u, v ∈Ω, then vu ∈ τ ⇒ v ∈ τ , so τ contains all of the ancestors of
its nodes;
A TYPED BRANCHING DIFFUSION 25
(iii) for all u ∈ τ , there exists Au ∈ {0,1,2, . . .} such that for j ∈N, uj ∈ τ
if and only if 1≤ j ≤ 1 +Au.
The set of all such trees is T, and we will use the symbol τ for a particular
tree. As our work concerns branching diffusions we shall often refer to the
labels of τ as particles. Note that for the binary branching mechanism in this
paper, P (Au = 1) ≡ 1; of course, here there is only one τ ∈ T—the binary
tree.
A Galton–Watson tree by itself only records the family structure of the
individuals, so to each individual u ∈ τ we give a mark (Xu, Yu, σu) which
contains the following information:
• σu ∈ [0,∞) is the lifetime of particle u, which also determines the fission
time of the particle as Su :=
v≤u σv . We may also refer to the Su as
death times;
• the function Xu(t) : [Su − σu, Su)→R describes the particle’s spatial mo-
tion in R during its lifetime;
• the function Yu(t) : [Su− σu, Su)→R describes the evolution of the parti-
cle’s type in R during its lifetime.
For clarity we must decide whether or not a particle is in existence at its
death time: our convention will be that a particle dies “infinitesimally be-
fore” its death time—this is why Xu and Yu are defined on [Su−σu, Su) and
not [Su−σu, Su]—so that at time Su the particle u has disappeared and has
been replaced by its two children.
We denote a particular marked tree by (τ,X,Y,σ), or the abbreviation
(τ,M), and the set of all marked Galton–Watson trees by T . For each
(τ,X,Y,σ) ∈ T , the set of particles alive at time t is defined as Nt := {u ∈
τ :Su − σu ≤ t < Su}. For any given marked tree (τ,M) ∈ T we can distin-
guish individual lines of descent from the initial ancestor: ∅, u1, u2, u3, . . . ∈
τ , where ui is a child of ui−1 for all i ∈ {2,3, . . .} and u1 is a child of the
initial individual ∅. We call such a line of descent a spine and denote it by
ξ. In a slight abuse of notation we refer to ξt as the unique node in ξ that is
alive at time t, and also for the position of the particle that makes up the
spine at time t; that is, ξt :=Xu(t), where u ∈ ξ∩Nt. However, although the
interpretation of ξt should always be clear from the context, we introduce
the following notation for use where some ambiguity may still arise:
• nodet((τ,M, ξ)) := u if u ∈ ξ is the node in the spine alive at time t.
It is natural to think of the spine as a single diffusing particle ξt, or, strictly
speaking, the pair (ξt, ηt), where ηt is the type of the spine at time t.
We define nt to be a counting function that tells us which generation of
the spine is currently alive, or equivalently the number of fission times there
have been on the spine:
nt = |nodet(ξ)|.
26 Y. GIT, J. W. HARRIS AND S. C. HARRIS
The collection of all marked trees with a distinguished spine is the space
T̃ on which our probability measures will eventually be defined, but first we
define four filtrations on this space that contain different levels of information
about the branching diffusion.
• Filtration (Ft)t≥0. We define a filtration of T̃ made up of the σ-algebras
Ft := σ((u,Xu, Yu, σu) :Su ≤ t;
(u,Xu(s), Yu(s) : s ∈ [Su − σu, t]) : t ∈ [Su − σu, Su)),
which means that Ft is generated by the information concerning all par-
ticles that have lived and died before time t, and also those that are
still alive at time t. Each of these σ-algebras is a subset of the limit
F∞ := σ(
t≥0Ft).
• Filtration (F̃t)t≥0. We define the filtration (F̃t)t≥0 by augmenting the
filtration Ft with the knowledge of which node is the spine at time t; that
is, (F̃t)t≥0 := σ(Ft,nodet(ξ)) and F̃∞ := σ(
t≥0 F̃t), so that this filtration
knows everything about the branching diffusion and everything about the
spine.
• Filtration (Gt)t≥0. (Gt)t≥0 is a filtration of T̃ defined by Gt := σ(ξs : 0 ≤
s≤ t), and G∞ := σ(
t≥0 Gt). These σ-algebras are generated only by the
spine’s motion and so do not contain the information about which nodes
of the tree τ make up the spine.
• Filtration (G̃t)t≥0. As we did in going from Ft to F̃t we create (G̃t)t≥0
from (Gt)t≥0 by including knowledge of which nodes make up the spine:
(G̃t)t≥0 := σ(Gt,nodet(ξ)) and G̃∞ := σ(
t≥0 G̃t). This means that G̃t also
knows when the fission times on the spine occurred, whereas Gt does not.
Now that we have defined the underlying space and filtrations, we can de-
fine the probability measures of interest. We let the typed branching diffusion
be as described in Section 1.1, with the probability measures {P x,y :x, y ∈R}
on (T̃ ,F∞) representing the law of this typed branching diffusion when ini-
tially started with a single particle at (x, y).
We recall from [18] that, if f is an F̃t-measurable function, we can write
fu1{ξt=u},(44)
where fu is Ft-measurable. Now we can extend P x,y to a measure P̃ x,y on
(T̃ , F̃∞) by choosing the particle that continues the spine uniformly each
time there is a birth on the spine; more precisely, for any f ∈ mF̃t with
representation like (44), we have:
f dP̃ x,y(τ,M, ξ) :=
dP x,y(τ,M).
A TYPED BRANCHING DIFFUSION 27
We construct the F̃t-measurable martingale ζ̃(t) as
ζ̃(t) := v+λ (ηt)e
{R(ηs)+1/2λ2A(ηs)}ds−E+λ t × 2nte−
R(ηs)ds
× eλξt−1/2λ
A(ηs)ds(45)
= v+λ (ηt)2
nteλξt−E
Observe that this is a product of single-particle martingales, details of which
can be found in [17] or [10]. One can think of these as h-transforms of the P̃ -
law of the spine: the first makes η an outward-drifting Ornstein–Uhlenbeck
process with drift parameter µλ; the second increases the breeding rate on
the spine to 2R(·); and the third adds a spatial drift to ξ.
Using the martingale ζ̃(t) we may define a measure Q̃
λ on (T̃ , F̃∞) by
dP̃ x,y
ζ̃(t)
ζ̃(0)
v+λ (y)
v+λ (ηt)2
nteλξt−E
t.(46)
And since ζ̃(t) is a product of h-transforms, under Q̃
λ the process may be
re-constructed path-wise according to the following description:
• starting from spatial position x and type y the spine (ξt, ηt) diffuses spa-
tially as a Brownian motion with infinitesimal variance A(ηt) and infinites-
imal drift λA(ηt);
• the type of the spine, ηt, begins at y and moves in type space as an
outward-drifting Ornstein–Uhlenbeck process with generator
+ µλy
• the spine branches at rate 2R(ηt), producing 2 particles;
• one of these particles is selected uniformly at random;
• the chosen offspring repeats stochastically the behavior of its parent;
• the other offspring particle initiates a P ·,·-BBM from its birth position
and type.
The change of measure (46) projects onto the sub-algebra Ft as a condi-
tional expectation:
dP̃ x,y
v+λ (y)
P̃ x,y(v+λ (ηt)2
nteλξt−E
t|Ft),
and it is a short calculation using the methods of, for example, Hardy and
Harris [10] to show that:
28 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Theorem 11. If we define Q
λ := Q̃
λ |F∞ , then Q
λ is a measure on
F∞ that satisfies
dP x,y
= Ẑ+λ (t) :=
Z+λ (t)
Z+λ (0)
Moreover under Q
λ , the path-wise construction of the branching diffusion
is the same as under Q̃λ.
Although the path-wise construction of the branching diffusion is the same
under Q
λ and Q̃
λ , only the measure Q̃
λ “knows” about the spine. It is
clear, however, that we have Q̃
λ (A) =Q
λ (A) for any A ∈ F∞.
Under the measure Q̃
λ only the behavior of the spine is altered, and
combining this observation with conditioning on the spine’s path and fission-
times gives us a very useful representation for Z+λ (t) under Q̃
λ that we shall
refer to as the spine decomposition:
λ (t)|G̃∞) =
v+λ (ηSu)e
λξSu−E
Su + v+λ (ηt)e
λξt−E+λ t.(47)
Throughout the rest of this article we will refer to the two pieces of this de-
composition as the “sum term” and the “spine term.” This decomposition
is discussed in detail for a wide variety of branching diffusions in [9], but
to derive it we simply note that the contributions to Z+λ (t) from the sub-
trees that branch off the spine have constant Q̃
λ -expectation because they
behave as if under the original measure P , and we know that Z+λ (t) is a P -
martingale. The spine decomposition reduces many calculations about the
behavior of Z+λ (t) under Q̃
λ to one-particle calculations about the spine,
and this observation is exploited in the spine proofs of Lp-bounds for some
families of additive martingales in [9].
7. Proof of Theorem 7. The short climb probability. With the spine
foundations firmly established in Section 6, we may proceed with the proof
of the short climb probability lower bound from Theorem 7.
First, recall definitions (38) and (39), where A
t is the event that there
exists a particle that makes the short climb along optimal path (x̄, ȳ), and
t (ξ) is the event that the spine makes the short climb. Note that ε controls
the proximity to x̄ and δ the proximity to ȳ. Importantly, we will only be
interested in taking λ= λ̄ throughout this section, although we will usually
just write λ for notational simplicity. Also recall throughout that t and τ
are related through (θ/(2µλ)) exp(2µλτ) = κ
A TYPED BRANCHING DIFFUSION 29
Proof of Theorem 7. The key step in the proof of this is the following
use of the spine change of measure: for any function g :R+ →R+ we have
P x,y(A
t ) =Q
Ẑ+λ (τ)
Ẑ+λ (τ)
≥ Q̃x,yλ
Ẑ+λ (τ)
≥ Q̃x,yλ
Ẑ+λ (τ)
; sup
s∈[0,τ ]
Ẑ+λ (s)≤ g(τ)
≥ g(τ)−1Q̃x,yλ
t (ξ); sup
s∈[0,τ ]
Ẑ+λ (s)≤ g(τ)
Essentially we just have to make the “correct” choice for both λ and g in
expression (48), although there will still remain a number of technicalities
to resolve.
The first idea is to ensure the (originally rare) event A
t actually occurs
under the new measure Q̃
λ by making the spine follow close to the required
path (x̄, ȳ); this is achieved by choosing the optimal value λ̄ for λ and choos-
ing τ to be on the natural time scale it would take the spine to reach position
t. In particular, this choice will mean that in the first line of the above set
of inequalities there is no significant loss of mass when replacing the event
t with A
t (ξ). Next, we wish to choose the smallest possible g that will
still leave some positive probability on the last line of the above argument.
Hence, we wish to identify the rate of growth of the martingale Z+λ under
λ , and this will essentially be governed by the contribution from the spine
itself.
With this is mind, and recalling the various properties of the optimal
paths and parameters from Section 4, for ε0 > 0 we define
gε0(τ) := exp
ψ+λ +
e2µλτ − (ψ+λ y
2 + λx)
and recall from (35) that the scaling between t and τ is fixed throughout,
where κ2t= (θ/(2µλ̄))e
2µλ̄τ for large t, hence t+ τ ∼ t. Note that since we
are only considering the optimal value λ= λ̄, we have
e2µλ̄τ = (κ2ψ+
− λ̄β)t=Θ(β,κ).
Then from (48) we have
P x,y(A
t )≥ gε0(τ)−1Q̃
t (ξ); sup
s∈[0,τ ]
Ẑ+λ (s)≤ gε0(τ)
.(49)
Our strategy for the rest of this proof is to show that the Q̃
λ -probability in
(49) is at least some ε′ > 0 for all sufficiently large t, uniformly for y ∈ [y0, y1],
30 Y. GIT, J. W. HARRIS AND S. C. HARRIS
so that the decay rate part of (49) matches the desired rate in the statement
of the theorem.
Conditioning on the spine’s path and birth times, G̃∞, and then making
use of some standard properties of conditional expectation we have
t (ξ); sup
s∈[0,τ ]
Ẑ+λ (s)≤ gε0(τ)
t (ξ); sup
s∈[0,τ ]
Ẑ+λ (s)≤ gε0(τ)
s∈[0,τ ]
Ẑ+λ (s)≤ gε0(τ)
since A
t (ξ) is G̃∞-measurable. We next observe that, conditional on G̃∞,
we can write Ẑ+λ (t) as
Ẑ+λ (t) = e
y2+λx)
λ (t− Su) + f(t)
,(50)
where the Z
λ are independent copies of Z
λ started from a single particle
at (ξSu , ηSu); and f(t) is the contribution to Z
λ (t) from the spine, which,
conditional on G̃∞, is a known function of t. Now if we could show, for
0< ε̃0 < ε0,
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
and sup
s∈[0,τ ]
(Ẑ+λ (s)− f̂(s))≤
gε0(τ)
where f̂(t) := e−(ψ
y2+λx)f(t), we would have sups∈[0,τ ] Ẑ
λ (s)≤ gε0(τ). Hence,
defining Ẑ+λ (s) := Ẑ
λ (s)− f̂(s), we have
s∈[0,τ ]
Ẑ+λ (s)≤ gε0(τ)
≥ Q̃x,yλ
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
; sup
s∈[0,τ ]
Ẑ+λ (s)≤
gε0(τ)
1{sups∈[0,τ ] f̂(s)≤gε̃0 (τ)/2}
s∈[0,τ ]
Ẑ+λ (s)≤
gε0(τ)
since, conditional on G̃∞, the supremum of f̂ on [0, τ ] is known.
We see from (50) that, conditional on G̃∞, Ẑ+λ (t) is a submartingale. This
is because the Q̃
λ -conditional expectation of each of the Z
λ in the sum
y2+λx)
λ (t− Su)(51)
A TYPED BRANCHING DIFFUSION 31
is constant, so the expectation of the sum cannot decrease, and in fact this
expectation increases every time there is a birth on the spine. Then by
Doob’s submartingale inequality we have
s∈[0,τ ]
Ẑ+λ (s)≤
gε0(τ)
= 1− Q̃x,yλ
s∈[0,τ ]
Ẑ+λ (s)≥
gε0(τ)
≥ 1− 2
gε0(τ)
λ (Ẑ
λ (τ)|G̃∞).
We must note here that the expectation on the above line is not a priori
finite. However, the expectation of each term in the sum (51) is bounded by
sups∈[0,τ ] f̂(s), which we have control over via an indicator function and so
we do not have to worry about this expectation blowing up.
So we need to show that for all sufficiently large τ and all y ∈ [y0, y1],
(ξ)∩{sups∈[0,τ ] f̂(s)≤gε̃0 (τ)/2}
gε0(τ)
λ (Ẑ
λ (τ)|G̃∞)
> ε′,
and hence also
t (ξ); sup
s∈[0,τ ]
Ẑ+λ (s)≤ gε0(τ)
as required. This will follow by combining both parts of the following result.
Lemma 12. Fix y1 > y0 > 0 and ε0 > ε̃0 > 0.
(i) For all sufficiently small ε, δ > 0, there exists some ε′ > 0 and T̃ > 0
such that for all y ∈ [y0, y1] and all t > T̃ ,
t (ξ); sup
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
> ε′.
(ii) As τ →∞,
gε0(τ)
λ (Ẑ
λ (τ)|G̃∞); sup
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
uniformly over y ∈ [y0, y1].
Then we have shown that, for any ε0 > 0, y1 > y0 > 0, and sufficiently
small ε, δ > 0, there exists a T > 0 such that, for all y ∈ [y0, y1] and all t > T ,
t−1 logP x,y(Aε,δt )≥−(Θ(β,κ) + ε0).
32 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Finally, we observe that the probability P x,y(A
t ) is trivially monotone
increasing in both ε and δ, and so it follows that if the result is true for all
sufficiently small ε and δ, it is in fact true for all ε, δ > 0. This completes
the proof of Theorem 7. �
Proof of Lemma 12(i). We will prove Lemma 12(i) in a sequence of
other lemmas, using a convenient coupling for the spine’s type process.
First recall that, under Q̃
λ , ηs solves the SDE
dηs =
θ dBs + µληs ds,
where Bs is a Q̃λ-Brownian motion. Noting that d(e
−µλsηs) = e
θ dBs,
we can construct e−µλsηs as a time-change of a Brownian motion with
e−µλsηs − η0 =
e−µλw dBw =
B̃(1− e−2µλs),
where B̃ is also a Q̃
λ -Brownian motion started at the origin.
In this way, we will construct ηy under P from a Brownian motion By
started at y
2µλ/θ where, for s ∈ [0,∞),
ηy(s) =
eµλsBy(1− e−2µλs).
To construct simultaneously all type processes ηy under the same measure
P, we first construct the process By0 as an independent Brownian motion
started at y0
2µλ/θ. Second, we construct the process B
y1 by running an
independent Brownian motion started at y1
2µλ/θ until it first hits the
path of By0 , at which point we couple the two processes together. Next, for
any other y ∈ (y0, y1), we run an independent Brownian motion By until it
first meets with either the process By0 below or By1 above, at which point
we couple it to the process it first hits.
Finally, we construct all the corresponding spatial processes ξy under P
from a single Brownian motion W by defining
ξy(s) =W
ηy(w)2 dw
ηy(w)2 dw,(52)
where W is started at x and is independent of the By processes.
Constructed in this way, for each y ∈ [y0, y1], the P-law of (ξy, ηy) is the
same as the Q̃
λ -law of (ξ, η).
Fixing µ ∈ (0,1) and K >max{y1,1}, we define the events and stopping
times
Ayε :=
By(s) ∈
,∀s ∈ (1− µ,1]
T0 := inf{t :By0(t) = 0}, TK := inf{t :By1(t) =K},
Ãε,K :=A
ε ∩Ay1ε ∩ {T0 > 1} ∩ {TK > 1}.
A TYPED BRANCHING DIFFUSION 33
Then, clearly P(Ãε,K)> 0 and, on the event Ãε,K , the coupling gives
0< ηy0(s)≤ ηy(s)≤ ηy1(s)≤K
eµλs,
for all s≥ 0 and y ∈ [y0, y1]. Note that our construction also ensures that if
event Ay0ε ∩Ay1ε occurs then so must Ayε for any y ∈ [y0, y1], hence Ayε ⊃ Ãε,K .
Lemma 13. Let ε > 0. On event Ãε,K , there exists a deterministic time
s0 = s0(ε)> 0 such that for all τ > s0,
s∈[0,τ ]
|ηy(s)− ȳ(s)| ≤ ε
for all y ∈ [y0, y1].
Proof. Set s1 =− 12µλ logµ and then, on event Ãε,K , for all τ ≥ s > s1
we have
ηy(s)−
eµλs,
for all y ∈ [y0, y1]. Writing
ȳ(s) =
1− e−2µλs
1− e−2µλτ
eµλs,
we see that there exists s2 = s2(ε)> 0 such that, for τ ≥ s > s2,
ȳ(s)−
eµλs.
Taking s3(ε) = max{s1, s2(ε)} now yields
|ηy(s)− ȳ(s)|< ε
eµλs ≤ ε
t(53)
for all τ ≥ s > s3 and all y ∈ [y0, y1].
Now consider s ∈ [0, s3]. On Ãε,K we have
|ηy(s)− ȳ(s)| ≤
eµλs3(1 +K),
and hence for some s4(ε)> 0 we have |ηy(s)− ȳ(s)| ≤ ε
t for all τ > s4, all
s ∈ [0, s3], and all y ∈ [y0, y1]. Taking s0(ε) =max{s3, s4} yields the result.
34 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Lemma 14. Let δ > 0. Then for all sufficiently small ε, there exists a
deterministic τ0 = τ0(ε, δ)> 0 such that, on Ãε,K , we have
s∈[0,τ ]
ηy(w)2 dw−
ȳ(w)2 dw
< δt(54)
for all τ > τ0 and all y ∈ [y0, y1].
Proof. Given any δ > 0, we first fix an ε > 0 sufficiently small such
that ε(2+ ε
; this yields a corresponding s3 = s3(ε), which is chosen
as at equation (53). Given this s3, we find τ1 = τ1(ε, δ)> 0 such that, for all
τ > τ1,
(K2 + 1)
e2µλw dw <
We now set τ0 = τ0(ε, δ) = max{s3, τ1}. With this choice of ε and τ0, we
proceed to show that the inequality (54) is satisfied. Note that τ0 is deter-
ministic and independent of y.
From equation (53) we see that, on Ãε,K and for s > s3,
ηy(w)2 dw ≥
ηy(w)2 dw+
ȳ(w)− ε
ȳ(w)2 dw−
ȳ(w)2 dw− 2
e2µλw dw
ȳ(w)2 dw−
e2µλw dw− (2ε) κt
ȳ(w)2 dw− δ
for all τ > τ0 and all y ∈ [y0, y1]. Similarly
ηy(w)2 dw ≤
ηy(w)2 dw+
ȳ(w) +
ȳ(w)2 dw+
ȳ(w)2 dw+K2
e2µλw dw+ ε
ȳ(w)2 dw+
A TYPED BRANCHING DIFFUSION 35
for all τ > τ0 and all y ∈ [y0, y1]. Finally, for s ∈ [0, s3], on Ãε,K we have
ηy(w)2 dw−
ȳ(w)2 dw
ηy(w)2 dw+
ȳ(w)2 dw
≤ (K2 + 1)
e2µλw dw < δt
for all τ > τ0 and all y ∈ [y0, y1]. �
Lemma 15. Let δ > 0. Then for all sufficiently small ε > 0, there exists
P-almost everywhere on Ãε,K a random time S0 = S0(δ, ε)<∞ such that
s∈[0,τ ]
ξy(s)− λa
ηy(w)2 dw
< δt,
for all y ∈ [y0, y1] and all τ > S0.
Proof. Given δ > 0, choose any δ′, δ′′ > 0 such that δ′(|β/λ|+ δ′′)< δ.
Recalling the construction of ξy at (52), we see from standard properties of
Brownian motion that there almost surely exists some S1 = S1(δ
′)<∞ such
s∈[0,t]
|W (s)|< δ′ for all t > S1.
s∈[0,τ ]
ηy(w)2 dw
ηy(w)2 dw
for all τ such that a
y(w)2 dw > S1, and by the coupling construction,
on Ãε,K this is true for all y ∈ [y0, y1] if a
y0(w)2 dw > S1. Then there
exists (P-almost everywhere on Ãε,K) a random time S2 = S2(δ
′)<∞, which
depends on By0 and S1, such that a
y(w)2 dw > S1 for all y ∈ [y0, y1] when
τ > S2.
Now by Lemma 14, given δ′′ and a sufficiently small ε, there exists a
deterministic τ0 = τ0(ε, δ
′′)> 0 such that, on Ãε,K ,
ηy(w)2 dw ≤ a
ȳ(s)2 ds+ δ′′t=
+ δ′′
t(56)
for all τ > τ0 and all y ∈ [y0, y1]. Combining the inequalities at (55) and (56),
we now see that, for τ > S0 = S0(ε, δ
′, δ′′) = max{S2, τ0},
s∈[0,τ ]
ξy(s)− λa
ηy(w)2 dw
= sup
s∈[0,τ ]
ηy(w)2 dw
for all y ∈ [y0, y1]. �
36 Y. GIT, J. W. HARRIS AND S. C. HARRIS
On combining Lemmas 14 and 15 and recalling the definition of optimal
path x̄ at (37), we obtain the following:
Lemma 16. Let δ > 0. Then for all sufficiently small ε > 0, there exists
P-almost everywhere on Ãε,K a random time S̃0 = S̃0(δ, ε)<∞ such that
s∈[0,τ ]
|ξy(s)− x̄(s)|< δt,
for all y ∈ [y0, y1], and all τ > S̃0.
We may now draw everything together to finish the proof of Lemma 12(i).
First we observe that since λ < 0, on event A
t (ξ),
s∈[0,τ ]
y2+λx) exp(ψ+λ η
s + λξs −E+λ s)
≤ e−(ψ
y2+λx) exp(ψ+λ (κ+ ε)
2t+ λ(−β − δ)t),
and so, given ε̃0, we can choose first δ and then ε sufficiently small so that
t (ξ)⊂
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
and, from Lemmas 13 and 16, there exists a random time T̃ = T̃ (δ, ε) <∞
such that on Ãε,K we have
s∈[0,τ ]
|ηy(s)− ȳ(s)|< ε
s∈[0,τ ]
|ξy(s)− x̄(s)|< δt
for all τ > T̃ and all y ∈ [y0, y1]. That is, Ãε,K ∩ {T̃ < τ} ⊂Aε,δt (ξy) for each
y ∈ [y0, y1], with the slight abuse of notation that
s∈[0,τ(t)]
|ηy(s)− ȳ(s)|< ε
t; sup
s∈[0,τ(t)]
|ξy(s)− x̄(s)|< δt
Note also that P(Ãε,K)> ε
′ for some ε′ > 0.
Combining the above, for any y ∈ [y0, y1] we have
t (ξ); sup
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
t (ξ)) = P(A
≥ P(Ãε,K; T̃ < τ)→ P(Ãε,K)
as τ →∞, as required. �
A TYPED BRANCHING DIFFUSION 37
Proof of Lemma 12(ii). Consider the expectation of the “sum term.”
We have
λ (Ẑ
λ (τ)|G̃∞) = e
y2+λx)
λ (t− Su)
= e−(ψ
y2+λx)
λ (t− Su)|G̃∞)
≤ e−(ψ
y2+λx)nτ max{eψ
η(Su)
2+λξ(Su)−E+λ Su :u < ξτ}
≤ nτ sup
s∈[0,τ ]
f̂(s).
Hence
gε0(τ)
λ (Ẑ
λ (τ)|G̃∞); sup
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
≤ Q̃x,yλ
gε̃0(τ)
gε0(τ)
; sup
s∈[0,τ ]
f̂(s)≤ gε̃0(τ)
≤ e−(ε0−ε̃0)tQ̃x,yλ (nτ ),
and we can now calculate Q̃
λ (nτ ) = Q̃
λ (Q̃
λ (nτ |G∞)), where G∞ the
σ-algebra generated by the path of the spine (not including the birth times).
Conditional on G∞, nτ is a Poisson random variable with mean given by
0 2(rη
s + ρ)ds, and using Fubini’s theorem we have
2(rη2s + ρ)ds
s)ds+2ρτ
e2µλτ −
− rθτ
+ 2ρτ
2y2κµλ
t+ o(τ).
So the Q̃
λ -expectation of nτ only grows linearly in t. Then since ε0− ε̃0 > 0,
the expression at (57) tends to 0 as t→∞. Moreover, the expectation at (57)
is bounded by the Q̃
λ -expectation, and hence the convergence is uniform
over y ∈ [y0, y1], as claimed. �
8. Martingale results. In this section we recall some existing and prove
some new martingale results that are intermediate steps in the proofs of
38 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Theorem 1 and the upper bound of Theorem 3. We recall from [13] that E−λ
[also written E−(λ)] and ∆(γ) are Legendre conjugates with
∆(γ) = inf
{E−(λ) + λγ}, E−(λ) = sup
{∆(γ)− γλ}.(58)
If, for λmin <λ< 0, we write γλ for the γ value which achieves the supremum
on the right-hand side of equation (58), then the functions λ 7→ γλ from
(−λmin,0) to (0,∞), and γ 7→ λγ from (0,∞) to (−λmin,0) are inverses of
each other and, of course, λγ is the λ value which achieves the infimum on
the left-hand side of equation (58). In addition, we note that
γλ =−
E−(λ) =
θa2λ2
θ− 8r− 4aλ2 ,(59)
that E−(λ) and ∆(γ) are convex functions, and that
c̃(θ) = sup{γ :∆(γ)> 0}= inf{−E−(λ)/λ :λmin <λ< 0}
= inf{c−λ :λmin < λ< 0}= c
λ̃(θ)
where
c−λ :=−E
λ /λ and
λ̃(θ) :=−
2(θ− 8r)(θρ+ 2ρ2 + rθ)
a(θ+4ρ)2
∈ (λmin,0).
A formula for c̃(θ) is given in equation (9). The following fundamental con-
vergence result for the Z−λ martingale was first partly proved in [13], but
also see [9] for a more complete proof using “spine” techniques.
Theorem 17. Suppose λ ∈ (λmin,0].
(i) If λ ∈ (λ̃(θ),0], the martingale Z−λ is uniformly integrable and has
an almost sure strictly positive limit.
(ii) If λ≤ λ̃(θ), then Z−λ (∞) = 0 almost surely.
The following convergence result was proved in [12] using martingales
based on Hermite polynomials.
Theorem 18. Let λ ∈ (λ̃(θ),0] and α< 1/4. For each P x,y starting law
and every continuous bounded function f :R 7→R, we have
f(Yu(t))e
αYu(t)
2+λ(Xu(t)+c
λ (∞),
A TYPED BRANCHING DIFFUSION 39
where
f0 :=
)1/4 ∫
f(y)eαy
y2φ(y)dy(62)
and φ(y) is the standard normal density.
In this paper, we require a corollary to this theorem which specifies more
precisely which particles contribute to the final limit.
Corollary 19. Let λ ∈ (λ̃(θ),0] and α < 1/4. For each P x,y starting
law and every continuous bounded function f :R 7→ R, we have for every
ε > 0
f(Yu(t)) e
αYu(t)
2+λXu(t)−E−λ t
1{|Xu(t)/t+γλ|<ε}−→a.s. f0Z
λ (∞)(63)
where γλ =− ∂∂λE
−(λ) and f0 is given at equation (62).
This last result will enable us to show in Section 10 that the almost sure
growth rate is at least as large as the expected growth rate, D(γ)≥∆(γ). It
is easy to see from Corollary 19 that when Z−λ (∞)> 0, there must exist at
least one particle near to −γλt in space. Further, because of the decay rate
of each term in the sum over particles at equation (63), it will be relatively
straightforward to improve this to get the required exponential numbers of
particles, exp(∆(γ)t), near −γλt for large times [as long as Z−λ (∞)> 0].
The following result concerns the rate at which the martingales Z+λ and
Z−λ converge to zero.
Theorem 20. Let λ ∈ (λmin,0). For every starting law, P x,y,
logZ±λ (t)
→ λ(c±λ − c
λ) a.s.
where c±λ is given at (5), and
c∗λ :=
c̃(θ), if λmin <λ≤ λ̃(θ),
c−λ , if λ̃(θ)≤ λ < 0.
Corollary 21. If λ ∈ (λmin,0), then Z+λ (t)→ 0 P x,y-almost surely.
The rate of convergence of the Z+λ martingale in part (i) of Theorem 20
is crucial in Section 9 to obtain the upper bound on the almost sure growth
rate, D(γ,κ) ≤ ∆(γ,κ). We also comment that if Corollary 19 were true
for all α < ψ+λ , then we could have gained this upper bound at that point.
40 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Although Corollary 19 is only proven for α < 1/4 (where we can utilize
suitable Hermite expansions), we conjecture that it holds for all α <ψ+λ .
Proof of Corollary 19. Let ε > 0 be small, µ := λ−ε, λ,µ ∈ (λ̃(θ),0),
f be a positive, continuous bounded function, α< 1/4 and note that γµ > γλ.
Then we have
f(Yu(t))e
αYu(t)
2+λXu(t)−E−λ t
1{Xu(t)<−γµt}
≤ e(E
µ −E−λ −εγµ)t
f(Yu(t))e
αYu(t)
2+µXu(t)−E−µ t
1{Xu(t)<−γµt}
f(Yu(t))e
αYu(t)
2+µXu(t)−E−µ t
−E−µ +(λ−µ)γµ)t.
Recall that E−(λ) is convex with ∂
E−(λ)≥ 0 and ∂
E−(λ) = γλ, so, from
the Taylor expansion,
E−λ −E
µ + (µ− λ)
E−(λ)
(µ− λ)2
E−(λ) + o((µ− λ)2).
Then taking ε > 0 small enough so that E−λ −E−µ +(λ−µ)γµ > 0, and using
Theorem 18, we find that for any δ > 0
limsup
f(Yu(t))e
αYu(t)
2+λXu(t)−E−λ t
1{Xu(t)<−(γλ+δ)t} = 0.
Similarly, we can show
limsup
f(Yu(t))e
αYu(t)
2+λXu(t)−E−λ t
1{Xu(t)>−(γλ−δ)t} = 0,
and hence the only contribution to the limit comes from the particles near
−γλt in space. Combining this with Theorem 18 we have
λ (∞) = limt→∞
f(Yu(t))e
αYu(t)
2+λXu(t)−E−λ t
= lim
f(Yu(t))e
αYu(t)
2+λXu(t)−E−λ t
1{|Xu(t)/t+γλ |<δ}.
Proof of Theorem 20. We use a useful technique brought to our
attention in [22]. Let p ∈ (0,1) so that, by Jensen’s inequality, Z±λ (t)p is a
supermartingale; then for u, v > 0 we have
(u+ v)p ≤ up + vp,
A TYPED BRANCHING DIFFUSION 41
and hence
Z±λ (t)
Yu(t)
2+λ(Xu(t)+c
Yu(t)
2+pλ(Xu(t)+c
For any ε > 0, Doob’s supermartingale inequality says
s≤w≤s+t
Z±λ (w)
p > εp
λ (s)
≤ ε−p
Yu(s)
2+pλ(Xu(s)+c
and then
s≤w≤s+t
eδwZ±λ (w)> ε
s≤w≤s+t
Z±λ (w)
p > e−pδ(s+t)εp
≤ ε−pepδt
Yu(s)
2+pλ(Xu(s)+c
p(λ(c±
)+δ)s
Now, if we can choose p ∈ (0,1) such that λ(c±λ − c
pλ) + δ < 0 and pψ
ψ+pλ, we must have e
δuZ±λ (u)→ 0 almost surely by using a familiar Borel–
Cantelli argument. [The condition pψ±λ < ψ
pλ guarantees that the expec-
tation in the last line above tends to a finite limiting value, hence stays
bounded over all times s, as can be checked by using formula (17), for ex-
ample.]
For all 0 ≤ p < 1 we find pψ±λ < ψ
pλ. Considering the graph of c
quickly see that, for λ ∈ [λ̃(θ),0), taking p as close to 1 as we like gives the
best rate. For λ ∈ [λmin, λ̃(θ)) we can choose p so that pλ= λ̃(θ), which gives
the best rate.
Recall from Theorem 17 that Z−λ (∞)> 0 when λ ∈ (λ̃(θ),0). Then, so far,
we have proved the following:
Lemma 22. For every starting law, P x,y, and for all ε > 0, if λ ∈ (λmin,0)
e−εt e−λ(c
)tZ±λ (t)→ 0 a.s.
42 Y. GIT, J. W. HARRIS AND S. C. HARRIS
where
c∗λ :=
c̃(θ), if λmin <λ≤ λ̃(θ),
c−λ , if λ̃(θ)≤ λ < 0.
It is clear that this gives the required upper bound of
lim sup
logZ±λ (t)
≤ λ(c±λ − c
Now, for any ε > 0, if λ ∈ (λmin, λ̃(θ)] then
Yu(t)
2+λ(Xu(t)+c̃(θ)t) ≥ eλ(Lt+c̃(θ)t)+εt →∞ a.s.
since we know that Lt := inf{Xu(t) : u ∈ N(t)} satisfies Lt/t→ −c̃(θ) a.s.
Otherwise, with λ∈ (λ̃(θ),0),
Yu(t)
2+λ(Xu(t)+c
t) ≥ eεtZ−λ (t)→∞ a.s.
since here Z−λ (∞)> 0 a.s. Thus, in all cases,
lim inf
logZ±λ (t)
≥ λ(c±λ − c
which completes the proof of Theorem 20. �
9. Proof of Theorem 3. Upper bound. The idea for the upper bound
proof is to overestimate indicator function by exponentials, and then re-
arrange the expressions to form martingale terms.
Simply observe that for λ ∈ (λmin,0),
Nt(γ, [κ
t,∞)) =
1{Xu(t)≤−γt;Yu(t)≥κ
1{Xu(t)≤−γt;Yu(t)2≥κ2t}e
(Yu(t)
2−κ2t)+λ(Xu(t)+γt)
≤ e(E
−κ2ψ+
+λγ)t
Yu(t)
2+λXu(t)−E+λ t
≤ e−λ(c
)tZ+λ (t)e
+λγ−κ2ψ+
where E±λ =−λc
Recall from equations (11) and (32) that E−λ + λγ − κ2ψ
λ has a minimal
value of ∆(γ,κ) achieved when λ= λ̄(γ,κ). Since c̃(θ) is the minimal value
of cλ, Theorem 20 implies that
lim sup
t−1 logZ+λ (t)≤ λ(c
λ − c
λ )(65)
A TYPED BRANCHING DIFFUSION 43
almost surely for all λ ∈ (λmin,0).
In cases where ∆(γ,κ)< 0, we can use the optimal value for λ, Theorem 20
and trivially note that Nt(γ, [κ
t,∞)) is integer valued to deduce that
1{Yu(t)≥κ
t;Xu(t)≤−γt} = 0
eventually, almost surely. Hence, D(γ,κ) =−∞ almost surely if ∆(γ,κ)< 0.
Otherwise we have ∆(γ,κ)≥ 0, which in fact guarantees that γ ∈ (0, c̃(θ)]
and hence λ̄(γ,κ) ∈ [λ̃(θ),0). Then since
lim sup
t−1 logNt(γ, [κ
t,∞))
≤ lim sup
t−1 log(e−λ(c
)tZ+λ (t)) + (E
λ + λγ − κ
2ψ+λ )
we can again make use of Theorem 20 and the minimizing λ value, λ̄(γ,κ),
to get the bound
limsup
t−1 logNt(γ, [κ
t,∞))≤∆(γ,κ) almost surely,
as desired.
Notice that, when ∆(γ,κ) = 0, the right-hand side of the inequality at
(64) will tend to infinity (see Corollary 19). Then, on the boundary, we have
only shown that lim sup t−1 logNt(γ, [κ
t,∞))≤ 0.
10. Proof of Theorem 1. The spatial growth rate. We first bound the
spatial growth rate above. Suppose that C ⊂ R is Borel-measurable with
y2φ(y)dy > 0. Let λ ∈ (λmin,0), then
1{Xu(t)≤−γt;Yu(t)∈C} ≤
1{Yu(t)∈C}e
λ(Xu(t)+γt)
= e(E
+λγ)t
1{Yu(t)∈C} e
λXu(t)−E−λ t
≤ e(E
+λγ)tZ−λ (t).
Recalling equations (8) and (19), we therefore have
1{Xu(t)≤−γt;Yu(t)∈C} ≤ e
∆(γ)tZ−λγ (t).
Now if γ ≥ c̃(θ), corresponding to λγ ∈ (λmin, λ̃(θ)] and having ∆(γ)≤ 0,
we know from Theorem 17 that Z−λγ (∞) = 0 almost surely. Then,
γ > c̃(θ) ⇒
1{Xu(t)≤−γt;Yu(t)∈C} = 0 eventually, a.s.
44 Y. GIT, J. W. HARRIS AND S. C. HARRIS
Otherwise, if γ ∈ (0, c̃(θ)), corresponding to λγ ∈ (λ̃(θ),0) and having ∆(γ)>
0, Theorem 17 tells us that Z−λγ (∞)> 0 almost surely, hence
lim sup
t−1 log
1{Xu(t)≤−γt;Yu(t)∈C} ≤∆(γ).
Now we bound the growth rate from below. Let ε > 0 be small, λ̃(θ) <
λ < 0, and µ = λ − ε. We recall now that E−λ is convex so
≥ 0 and
γµ > γλ. Then
eλXu(t)−E
1{−(γλ+ε)t≤Xu(t)≤−(γλ−ε)t;Yu(t)∈C}
eλ(−(γλ+ε)t)−E
1{−(γλ+ε)t≤Xu(t)≤−(γλ−ε)t;Yu(t)∈C}
= e(−λγλ−E
−λε)t ∑
1{−(γλ+ε)t≤Xu(t)≤−(γλ−ε)t;Yu(t)∈C}
≤ e(−λγλ−E
−λε)t ∑
1{Xu(t)≤−(γλ−ε)t;Yu(t)∈C}.
t−1 log
1{Yu(t)∈C}e
λXu(t)−E−λ t
1{|Xu(t)/t+γλ |<ε}
≤−λγλ−E−λ − λε+ t
−1 log
1{Xu(t)≤−(γλ−ε)t;Yu(t)∈C}.
Letting t→∞, using Corollary 19 and remembering that for λ̃(θ)< λ≤ 0
we have Z−λ (∞)> 0 a.s., we find
0≤−λγλ −E−λ − λε+ lim inft→∞ t
−1 log
1{Xu(t)≤−(γλ−ε)t;Yu(t)∈C}
and as ε > 0 can be arbitrarily small we have
lim inf
t−1 log
1{Xu(t)≤−γλt;Yu(t)∈C} ≥E
λ + λγλ.
Equivalently,
lim inf
t−1 log
1{Xu(t)≤−γt;Yu(t)∈C} ≥E
+ λγγ =∆(γ)
and hence the lim sup and lim inf agree as required.
We note that these proofs will easily adapt to cover a multi-type branching
Brownian motion where the types evolve as a finite state Markov chain,
such as found in [2], where it will also be possible to prove the analogous
A TYPED BRANCHING DIFFUSION 45
convergence theorem required when we have a finite type space by adapting
the proof of Theorem 18 found in [12].
In the standard branching Brownian motion case things are even simpler
to adapt (where, of course, there is no need for any convergence result akin to
Theorem 18). All the information necessary is contained in the martingales
u∈Nt exp(λXu(t)− (λ
2/2 + r)t) studied by Neveu [22] and, as first came
to our attention during discussions with J. Warren, the martingale with
parameter λ can only be capable of “counting” particles near γλt in space
at large times t, so when this martingale is uniformly integrable particles
must perpetually be found with the corresponding speed. Of course, in this
case more precise results, in the spirit of Watanabe [25], also exist.
Acknowledgments. We would like to thank two anonymous referees for
providing extremely helpful and thorough reviews of earlier incarnations
of this manuscript. Their numerous invaluable comments led to a much
improved presentation of this work.
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MR0237007
Y. Git
Statistical Laboratory
Cambridge University
22 Mill Street
Cambridge CB1 2HP
E-mail: Yoav.Git@gmail.com
J. W. Harris
Department of Mathematics
University of Bristol
University Walk
Bristol BS8 1TW
E-mail: john.harris@bristol.ac.uk
S. C. Harris
Department of Mathematical Sciences
University of Bath
Bath BA2 7AY
E-mail: s.c.harris@bath.ac.uk
URL: http://people.bath.ac.uk/massch/
http://www.ams.org/mathscinet-getitem?mr=1417979
http://www.ams.org/mathscinet-getitem?mr=1991122
http://www.ams.org/mathscinet-getitem?mr=0199891
http://www.ams.org/mathscinet-getitem?mr=1601737
http://www.ams.org/mathscinet-getitem?mr=2037473
http://www.ams.org/mathscinet-getitem?mr=1601749
http://www.ams.org/mathscinet-getitem?mr=1349164
http://www.ams.org/mathscinet-getitem?mr=0400428
http://www.ams.org/mathscinet-getitem?mr=1952568
http://www.ams.org/mathscinet-getitem?mr=1046418
http://www.ams.org/mathscinet-getitem?mr=1659492
http://www.ams.org/mathscinet-getitem?mr=0758258
http://www.ams.org/mathscinet-getitem?mr=0237007
mailto:Yoav.Git@gmail.com
mailto:john.harris@bristol.ac.uk
mailto:s.c.harris@bath.ac.uk
http://people.bath.ac.uk/massch/
	Introduction
	The branching model
	Application to reaction–diffusion equations
	Main results
	Martingales
	The asymptotic growth-rate of particles along spatial rays
	The asymptotic shape and growth of the branching diffusion
	Some expectation calculations
	The expected rate of growth along spatial rays
	The expected asymptotic shape
	Short climb large deviation heuristics
	A birth–death process
	Finding the optimal path and probability
	An important note on the optimal paths
	Proof of Theorem 3. Lower bound
	The ``spine'' setup and results
	Proof of Theorem 7. The short climb probability
	Martingale results
	Proof of Theorem 3. Upper bound
	Proof of Theorem 1. The spatial growth rate
	Acknowledgments
	References
	Author's addresses
