This article extends the results of Fang & Zeitouni (2012a) on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0,1] under a slight restriction. We find the asymptotics of the maximum up to an OP(1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein (2015b) when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang & Zeitouni (2012a) instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.
The more involved question of finding the second order of the maximum was first solved by @cite_15 0.3mm for the case @math and @math , and later by @cite_21 , when the law of the increments changes a finite number of times but is not necessarily Gaussian. In his proof, Mallein develops a time-inhomogeneous version of the spinal decomposition for the BRW. The argument presented in this paper was first developed, without the knowledge of Mallein's results, in Section 2.4 of @cite_11 and instead generalizes the approach of @cite_15 . The proof rely on the control of the increments of high points at every @math .
@cite_15 @cite_21 @cite_11
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We study the maximum of a Gaussian field on @math ( @math ) whose correlations decay logarithmically with the distance. Kahane Kah85 introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas DRSV12a DRSV12b extended Kahane's construction to the critical case and established the KPZ formula at criticality. Moreover, they made in DRSV12a several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in DRSV12a : we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale. Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. We revisit the discrete additive and multiplicative coalescents, starting with n particles with unit mass. These cases are known to be related to some “combinatorial coalescent processes”: a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process ((G(n,p))_p ) in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as (n + ). We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous (Ann Probab 25:812–854, 1997): we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time ( ), converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift ((B_t+ t - t^2 2, t 0) ), but we also construct a version of the standard augmented multiplicative coalescent of (Probab Theory Relat, 2013) using an additional Poisson point process. We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space-time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space-time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc.