We identify the fluctuations of the partition function for a class of random energy models, where the energies are given by the positions of the particles of the complex-valued branching Brownian motion (BBM). Specifically, we provide the weak limit theorems for the partition function in the so-called "glassy phase" -- the regime of parameters, where the behaviour of the partition function is governed by the extrema of BBM. We allow for arbitrary correlations between the real and imaginary parts of the energies. This extends the recent result of Madaule, Rhodes and Vargas, where the uncorrelated case was treated. In particular, our result covers the case of the real-valued BBM energy model at complex temperatures.
We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases (I-III) of this model. For the properly rescaled partition function, in Phase III and on the boundaries I/III and II/III, we prove a central limit theorem with a random variance. In Phase I and on the boundary I/II, we prove an a.s. and $L^1$ martingale convergence. All results are shown for any given correlation between the real and imaginary parts of the random energy.
We consider a model of Branching Brownian Motion in which the usual spatially-homogeneous and catalytic branching at a single point are simultaneously present. We establish the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.
In this article we establish the magnitude of fluctuations of the extreme particle in the model of binary branching Brownian motion with a single catalytic point at the origin.
We first study the convergence of solutions of a system of F-KPP equations related to irreducible multitype branching Brownian motions with Heaviside-type initial conditions to traveling wave solutions. Then we apply this convergence result to prove that the extremal processes of irreducible multitype branching Brownian motions converges weakly to a cluster point process.
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation pr…
Our purpose in this paper is to determine the limiting distribution and the evolution rate of particles near the frontier of branching Brownian motions. Here the branching rate is given by a Kato class measure with compact support in Euclidean space. Our investigation focuses on the two dimensional case.
Conditioning a branching Brownian motion to have an atypically low maximum leads to a suppression of the branching mechanism. In this note, we consider a branching Brownian motion conditioned to have a maximum below $\sqrt{2}αt$ ($α<1$). We study the precise effects of an early/late first branching time and a low/high first branching location under this condition. We do so by imposing additional constraints on the first branching time and location. We obtain large deviation estimates, as well as the optimal first branching time and location given the additional constraints.
Using one-dimensional branching Brownian motion in a periodic environment, we give probabilistic proofs of the asymptotics and uniqueness of pulsating travelling waves of the F-KPP equation in a periodic environment. This paper is a sequel to [Ren et al. Branching Brownian motion in a periodic environment and existence of pulsating travelling waves], in which we proved the existence of the pulsating travelling waves in the supercritical and critical cases using the limits of the additive and derivative martingales of branching Brownian motion in a periodic environment.
We consider the model of branching Brownian motion with a single catalytic point at the origin and binary branching. We establish some fine results for the asymptotic behaviour of the numbers of particles travelling at different speeds and give an explicit characterisation of the spatial distribution of particles travelling at the critical speed.
We study the height of the maximal particle at time $t$ of a one dimensional branching Brownian motion with a space-dependent branching rate. The branching rate is set to zero in finitely many intervals (obstacles) of order $t$. We obtain almost sure asymptotics of the first order of the maximum, describe the path of a particle reaching this height and describe its dependence on the size and location of the obstacles.
From multitype branching Brownian motions to branching Markov additive processes
University of Chinese Academy of Sciences（中国科学院大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；LMAM School of Mathematical Sciences & Center for Statistical Science, Peking University（北京大学数学学院（LMAM）及统计科学中心）；School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学科学学院）
We consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motions with drift $-\sqrt{2+2\varepsilon}$, undergo dyadic fission at rate $1$, and are killed when they reach the origin. We obtain a Yaglom type asymptotic result, showing that the long run expected number of particles conditioned on survival grows exponentially as $1/\sqrt{\varepsilon}$ as the process approaches criticality.
We determine the long time behavior and the exact order of the tail probability for the maximal displacement of a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of the associated Schrödinger type operator. To establish our results, we show a sharp and locally uniform growth order of the Feynman-Kac semigroup.
Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for resul…
We find the exponential growth rate of the population outside a ball with time dependent radius for a branching Brownian motion in Euclidean space. We then see that the upper bound of the particle range is determined by the principal eigenvalue of the Schrödinger type operator associated with the branching rate measure and branching mechanism. We assume that the branching rate measure is small enough at infinity, and can be singular with respect to the Lebesgue measure. We finally apply our results to several concrete models.
The branching Brownian sausage in $\mathbb{R}^d$ was defined by Engländer in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R}^d$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated exponentially shrinking branching Brownian sausage (BBM-sausage). Using a previous result on the density of the support of BBM, and some well-known resu…
We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $βδ_0(\cdot)$, where $δ_0(\cdot)$ is the Dirac delta function and $β$ is some positive constant. We show that the distribution of the rightmost particle centred about $\fracβ{2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [6] for the degenerate case of catalytic branching.
We construct a class of one-dimensional diffusion processes on the particles of branching Brownian motion that are symmetric with respect to the limits of random martingale measures. These measures are associated with the extended extremal process of branching Brownian motion and are supported on a Cantor-like set. The processes are obtained via a time-change of a standard one-dimensional reflected Brownian motion on $\mathbb{R}_+$ in terms of the associated positive continuous additive functionals. The processes introduced in this paper may be regarded as an analogue of the Liouville Brownian…
We show that in branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$, the law of $R_t^*$, the maximum distance of a particle from the origin at time $t$, converges as $t\to\infty$ to the law of a randomly shifted Gumbel random variable.
We consider a standard binary branching Brownian motion on the real line. It is known that the maximal position $M_t$ among all particles alive at time $t$, shifted by $m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t$ converges in law to a randomly shifted Gumbel variable. Derrida and Shi (2017) conjectured the precise asymptotic behaviour of the corresponding lower deviation probability $\mathbb{P}(M_t \leq \sqrt{2}αt)$ for $α< 1$. We verify their conjecture, and describe the law of the branching Brownian motion conditioned on having a small maximum.
In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $ξ= \left(ξ(x)\right)_{x\in\mathbb{R}}$ satisfying certain conditions. We show that the maximum position $M_t$ of particles alive at time $t$ satisfies a quenched strong law of large numbers and an annealed invariance principle.
We study the lower deviation probability of the position of the rightmost particle in a branching Brownian motion and obtain its large deviation function
Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion with negative drift, particles can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle's location. Under some assumptions, we obtain the limit in probability of the number of particles in any given interval and an explicit formula for the asymptotic empirical density of the fitness distr…
The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $σ_1^2=σ_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $σ_1^2=1\pm t^{-α}$ and $σ_2^2=1\pm t^{-α}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<α<\frac{1}{2}$. This is due to the localisati…
Fine asymptotics for the consistent maximal displacement of branching Brownian motion
众所周知，分支布朗运动中的最大粒子在时间$t$位于$\sqrt2 t - \frac{3}{2\sqrt2}\log t$附近。然后，人们可能会问边界附近粒子的路径：它们能离这条临界曲线多近？对这一问题已经形成了两种不同的方法。我们在每种情况下最知名的界限，揭示新的定性功能，包括显着的差异，这两种方法。
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
We study a dyadic branching Brownian motion on the real line with absorption at 0, drift $μ\in \mathbb{R}$ and started from a single particle at position $x>0.$ When $μ$ is large enough so that the process has a positive probability of survival, we consider $K(t),$ the number of individuals absorbed at 0 by time $t$ and for $s\ge 0$ the functions $ω_s(x):= \mathbb{E}^x[s^{K(\infty)}].$ We show that $ω_s<\infty$ if and only of $s\in[0,s_0]$ for some $s_0>1$ and we study the properties of these functions. Furthermore, for $s=0, ω(x) := ω_0(x) =\mathbb{P}^x(K(\infty)=0)$ is the cumulative distrib…
This work is a continuation of the manuscript "the structure of extreme level sets in branching Brownian motion", in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. Our main finding here is that most of the particles in an extreme level set come from only a small fraction of th…
We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.
