arXiv subjects

Julien Berestycki

Publications and source records attributed to Julien Berestycki.

At least 37 records · Page 2Linked to original sources

arXiv2017

Exact solution and precise asymptotics of a Fisher-KPP type front

The present work concerns a version of the Fisher-KPP equation where the nonlinear term is replaced by a saturation mechanism, yielding a free boundary problem with mixed conditions. Following an idea proposed in [BrunetDerrida.2015], we show that the Laplace transform of the initial condition is directly related to some functional of the front position $μ_t$. We then obtain precise asymptotics of the front position by means of singularity analysis. In particular, we recover the so-called Ebert and van Saarloos correction [EbertvanSaarloos.2000], we obtain an additional term of order $\log t /t$ in this expansion, and we give precise conditions on the initial condition for those terms to be present.

cond-mat.stat-mech↗

arXiv2016

A note of the convergence of the Fisher-KPP front centred around its $α$-level

We consider the solution $u(x,t)$ of the Fisher-KPP equation $\partial_t u=\partial_x^2u+u-u^2$ centred around its $α$-level $μ_t^{(α)}$ defined as $u(μ_t^{(α)},t)=α$. It is well known that for an initial datum that decreases fast enough, then $u(μ_t^{(α)}+x,t)$ converges as $t\to\infty$ to the critical travelling wave. We study in this paper the speed of this convergence and the asymptotic expansion of $μ_t^{(α)}$ for large~$t$. It is known from Bramson that for initial conditions that decay fast enough, one has $μ_t^{(α)}=2t-(3/2)\ln t+\text{Cste}+o(1)$. Work is under way \cite{nrr} to show that the $o(1)$ in the expansion is in fact a $k^{(α)}/\sqrt t+\mathcal O(t^{ε-1})$ for any $ε>0$ for some $k^{(α)}$, where it is not clear at this point whether $k^{(α)}$ depends or not on $α$. We show that, unless the time derivative of $μ_t^{(α)}$ has a very unexpected behaviour at infinity, the coefficient $k^{(α)}$ does not, in fact, depend on $α$. We also conjecture that, for an initial condition that decays fast enough, one has in fact $μ_t^{(α)}=2t-(3/2)\ln t+\text{Cste}-(3\sqrtπ)/\sqrt t+g (\ln t)/t +o (1/t)$ for some constant~$g$ which does not depend on $α$.

math.AP↗

arXiv2016

Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

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 distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption. We give three descriptions of the family $ω_s, s\in [0,s_0]$ through a single pair of functions, as the two extremal solutions of the Kolmogorov-Petrovskii-Piskunov (KPP) traveling wave equation on the half-line, through a martingale representation and as an explicit series expansion. We also obtain a precise result concerning the tail behavior of $K(\infty)$. In addition, in the regime where $K(\infty)>0$ almost surely, we show that $u(x,t) := \mathbb{P}^x(K(t)=0)$ suitably centered converges to the KPP critical travelling wave on the whole real line.

math.PR↗

arXiv2016

The number of accessible paths in the hypercube

Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube $\{0,1\}^L$ where each node carries an independent random variable uniformly distributed on $[0,1]$, except $(1,1,\ldots,1)$ which carries the value $1$ and $(0,0,\ldots,0)$ which carries the value $x\in[0,1]$. We study the number $Θ$ of paths from vertex $(0,0,\ldots,0)$ to the opposite vertex $(1,1,\ldots,1)$ along which the values on the nodes form an increasing sequence. We show that if the value on $(0,0,\ldots,0)$ is set to $x=X/L$ then $Θ/L$ converges in law as $L\to\infty$ to $\mathrm{e}^{-X}$ times the product of two standard independent exponential variables. As a first step in the analysis, we study the same question when the graph is that of a tree where the root has arity $L$, each node at level 1 has arity $L-1$, \ldots, and the nodes at level $L-1$ have only one offspring which are the leaves of the tree (all the leaves are assigned the value 1, the root the value $x\in[0,1]$).

math.PR↗

arXiv2015

Vanishing corrections for the position in a linear model of FKPP fronts

Take the linearised FKPP equation \[\partial_t h =\partial^2_x h +h\] with boundary condition $h(m(t),t)=0$. Depending on the behaviour of the initial condition $h_0(x)=h(x,0)$ we obtain the asymptotics - up to a $o(1)$ term $r(t)$ - of the absorbing boundary $m(t)$ such that $ω(x):=\lim_t h(x+m(t) ,t)$ exists and is non-trivial. In particular, as in Bramson's results for the non-linear FKPP equation, we recover the celebrated $-(3/2)\log t$ correction for initial conditions decaying faster than $x^νe^{-x}$ for some $ν<-2$. Furthermore, when we are in this regime, the main result of the present work is the identification (to first order) of the $r(t)$ term which ensures the fastest convergence to $ω(x)$. When $h_0(x)$ decays faster than $x^νe^{-x}$ for some $ν<-3$, we show that $r(t)$ must be chosen to be $-3\sqrt{π/t}$ which is precisely the term predicted heuristically by Ebert-van Saarloos in the non-linear case. When the initial condition decays as $x^νe^{-x}$ for some $ν\in [-3,-2)$, we show that even though we are still in the regime where Bramson's correction is $-(3/2)\log t$, the Ebert-van Saarloos correction has to be modified. Similar results were recently obtained by Henderson using an analytical approach and only for compactly supported initial conditions.

math.PR↗

arXiv2014

A small-time coupling between $Λ$-coalescents and branching processes

We describe a new general connection between $Λ$-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for $N^Λ(t)$ to the classical upper and lower indices arising in the study of pathwise properties of Lévy processes.

math.PR↗

arXiv2014

Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.

math.PR↗

arXiv2014

Accessibility percolation with backsteps

Consider a graph in which each site is endowed with a value called \emph{fitness}. A path in the graph is said to be "open" or "accessible" if the fitness values along that path is strictly increasing. We say that there is accessibility percolation between two sites when such a path between them exists. Motivated by the so called House-of-Cards model from evolutionary biology, we consider this question on the $L$-hypercube $\{0,1\}^L$ where the fitness values are independent random variables. We show that, in the large $L$ limit, the probability that an accessible path exists from an arbitrary starting point to the (random) fittest site is no more than $x^*_{1/2}= 1-\frac12\sinh^{-1}(2) =0.27818\ldots$ and we conjecture that this probability does converge to $x^*_{1/2}$. More precisely, there is a phase transition on the value of the fitness $x$ of the starting site: assuming that the fitnesses are uniform in $[0,1]$, we show that, in the large $L$ limit, there is almost surely no path to the fittest site if $x>x^*_{1/2}$ and we conjecture that there are almost surely many paths if $x<x^*_{1/2}$. If one conditions on the fittest site to be on the opposite corner of the starting site rather than being randomly chosen, the picture remains the same but with the critical point being now $x^*_1= 1-\sinh^{-1}(1)= 0.11863\ldots$. Along the way, we obtain a large $L$ estimation for the number of self-avoiding paths joining two opposite corners of the $L$-hypercube.

math.PR↗

arXiv2013

Critical branching Brownian motion with absorption: particle configurations

We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x > 0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-\sqrt{2}$, and particles are absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.

math.PR↗

arXiv2013

On Exceptional Times for generalized Fleming-Viot Processes with Mutations

If $\mathbf Y$ is a standard Fleming-Viot process with constant mutation rate (in the infinitely many sites model) then it is well known that for each $t>0$ the measure $\mathbf Y_t$ is purely atomic with infinitely many atoms. However, Schmuland proved that there is a critical value for the mutation rate under which almost surely there are exceptional times at which $\mathbf Y$ is a finite sum of weighted Dirac masses. In the present work we discuss the existence of such exceptional times for the generalized Fleming-Viot processes. In the case of Beta-Fleming-Viot processes with index $α\in\,]1,2[$ we show that - irrespectively of the mutation rate and $α$ - the number of atoms is almost surely always infinite. The proof combines a Pitman-Yor type representation with a disintegration formula, Lamperti's transformation for self-similar processes and covering results for Poisson point processes.

math.PR↗

arXiv2013

The genealogy of branching Brownian motion with absorption

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^3$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related model.

math.PR↗

arXiv2012

Critical branching Brownian motion with absorption: survival probability

We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $-\sqrt{2}$. Kesten (1978) showed that almost surely this process eventually dies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $t$. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007).

math.PR↗

arXiv2012

The $Λ$-coalescent speed of coming down from infinity

Consider a $Λ$-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit a deterministic function $v:(0,\infty)\to(0,\infty)$ such that $N_t/v(t)\to1$, almost surely, and in $L^p$ for any $p\geq1$, as $t\to0$. Our approach relies on a novel martingale technique.

math.PR↗

arXiv2012

Asymptotic sampling formulae for Lambda-coalescents

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Lambda-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to infinity. Some of our results hold in the case of a general Lambda-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at α=3/2, where α\in(1,2) is the exponent of regular variation.

math.PR↗

arXiv2011

Hitting properties and non-uniqueness for SDE driven by stable processes

We study a class of self-similar jump type SDEs driven by Hölder-continuous drift and noise coefficients. Using the Lamperti transformation for positive self-similar Markov processes we obtain a necessary and sufficient condition for almost sure extinction in finite time. We then show that for certain parameters pathwise uniqueness holds in a restricted sense, namely among solutions spending a Lebesgue-negligible amount of time at 0. A direct power transformation plays a key role.

math.PR↗

arXiv2010

The prolific backbone for supercritical superdiffusions

We develop an idea of Evans and O'Connell, Englander and Pinsky and Duquesne and Winkel by giving a pathwise construction of the so called `backbone' decomposition for supercritical superprocesses. Our results also complement a related result for critical $(1+β)$-superprocesses given in Etheridge and Williams \cite{EW}. Our approach takes an analytical point of view which is more in the spirit of the original Evans and O'Connell paper.

math.PR↗

arXiv2010

Survival of near-critical branching Brownian motion

Consider a system of particles performing branching Brownian motion with negative drift $μ= \sqrt{2 - ε}$ and killed upon hitting zero. Initially there is one particle at $x>0$. Kesten showed that the process survives with positive probability if and only if $ε>0$. Here we are interested in the asymptotics as $\eps\to 0$ of the survival probability $Q_μ(x)$. It is proved that if $L= π/\sqrtε$ then for all $x \in \R$, $\lim_{ε\to 0} Q_μ(L+x) = θ(x) \in (0,1)$ exists and is a travelling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when $x<L$ and $L-x \to \infty$. The proofs rely on probabilistic methods developed by the authors in a previous work. This completes earlier work by Harris, Harris and Kyprianou and confirms predictions made by Derrida and Simon, which were obtained using nonrigorous PDE methods.

math.PR↗

arXiv2008

Large deviations for Branching Processes in Random Environment

A branching process in random environment $(Z_n, n \in \N)$ is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of {\it large deviations}. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is smaller than the typical geometric growth rate $\bar L$ and $ Z_n \ge e^{cn}$ when $c > \bar L$. One way to obtain such an atypical rate of growth is to have a typical realization of the branching process in an atypical sequence of environments. This gives us a general lower bound for the rate of decrease of their probability. When each individual leaves at least one offspring in the next generation almost surely, we compute the exact rate function of these events and we show that conditionally on the large deviation event, the trajectory $t \mapsto \frac1n \log Z_{[nt]}, t\in [0,1]$ converges to a deterministic function $f_c :[0,1] \mapsto \R_+$ in probability in the sense of the uniform norm. The most interesting case is when $c < \bar L$ and we authorize individuals to have only one offspring in the next generation. In this situation, conditionally on $Z_n \le e^{cn}$, the population size stays fixed at 1 until a time $ \sim n t_c$. After time $n t_c$ an atypical sequence of environments let $Z_n$ grow with the appropriate rate ($\neq \bar L$) to reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0 on $[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].$

math.PR↗