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 total population stays roughly constant approximately $N$ particles. show that characteristic time scale for evolution this is order $(\log N)^{3}$, sense when measured these units, scaled number converges to variant Neveu's continuous-state process. Furthermore, genealogy then governed by coalescent process known as...

Coalescents with multiple collisions, also known as Λ-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation such a way several blocks can merge at same time to form single block. In case measure Λ is Beta (2−α, α) distribution, they are genealogies large populations where individual produce number offspring. Here, we use recent result Birkner et al. prove Beta-coalescents be embedded continuous...

For a finite measure $\varLambda$ on $[0,1]$, the $\varLambda$-coalescent is coalescent process such that, whenever there are $b$ clusters, each $k$-tuple of clusters merges into one at rate $\int_0^1x^{k-2}(1-x)^{b-k}\varLambda(\mathrm{d}x)$. It has recently been shown that if $1<\alpha<2$, in which $\operatorname {Beta}(2-\alpha,\alpha)$ distribution can be used to describe genealogy continuous-state branching (CSBP) with an $\alpha$-stable mechanism. Here we use facts about CSBPs...

We study $\lambda$-biased branching random walks on Bienaym\'e--Galton--Watson trees in discrete time. consider the maximal displacement at time $n$, $\max_{\vert u \vert =n} X(u) \vert$, and show that it almost surely grows a deterministic, linear speed. characterize this speed with help of large deviation rate function walk single particle. A similar result is given for minimal $\min_{\vert \vert$.

We consider a branching Brownian motion in Rd with d≥1 which the position Xt(u)∈Rd of particle u at time t can be encoded by its direction θt(u)∈Sd−1 and distance Rt(u) to 0. prove that extremal point process ∑δ (θt(u),Rt(u)−mt(d)) (where sum is over all particles alive mt(d) an explicit centering term) converges distribution randomly shifted, decorated Poisson on Sd−1×R. More precisely, so-called clan-leaders form Cox intensity proportional D∞(θ)e− 2rdrdθ, where D∞(θ) limit derivative...

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

In this paper we define and study self-similar ranked fragmentations. We first show that any fragmentation is the image of some partition-valued fragmentation, there in fact a one-to-one correspondence between laws these two types then give an explicit construction homogeneous fragmentations terms Poisson point processes. Finally use classical results on records processes to small-time behavior fragmentation.

We define and study a family of Markov processes with state space the compact set all partitions $N$ that we call exchangeable fragmentation-coalescence processes. They can be viewed as combination homogeneous fragmentation defined by Bertoin homogenous coalescence Pitman Schweinsberg or Möhle Sagitov. show they admit unique invariant probability measure some properties their paths equilibrium measure.

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 picked randomly. In this paper we give several results which belong to class {\it large deviations}. By contrast Galton-Watson case, here environments and can conspire achieve atypical events such as $Z_n \le e^{cn}$ when $c$ smaller than typical geometric growth rate $\bar L$ $ Z_n \ge $c &gt; \bar L$. One way obtain an have realization...

Motivated by an evolutionary biology question, we study the following problem: consider hypercube $\{0,1\}^L$ where each node carries independent random variable uniformly distributed on $[0,1]$, except $(1,1,\ldots,1)$ which value $1$ and $(0,0,\ldots,0)$ $x\in[0,1]$. We number $\Theta$ of paths from vertex to opposite along values nodes form increasing sequence. show that if is set $x=X/L$ then $\Theta/L$ converges in law as $L\to\infty$ $\mathrm{e}^{-X}$ times product two standard...

Nous présentons une méthode robuste qui permet de traduire des informations sur la vitesse descente l'infini d'un arbre généalogique en formules d'échantillonnages pour population sous-jacente. appliquons cette au cas où génélaogie est donnée par un $\varLambda$-coalescent. déduisons formule exacte le comportement asymptotique du spectre fréquences alléliques et nombre sites ségrégation, lorsque taille l'échantillon tend vers l'infini. Certains ces résultats sont valides dans général...

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

Take the linearised FKPP equation \[\partial_t h =\partial^2_x +h\] with boundary condition $h(m(t),t)=0$. Depending on behaviour of initial $h_0(x)=h(x,0)$ we obtain asymptotics - up to a $o(1)$ term $r(t)$ absorbing $m(t)$ such that $\omega(x):=\lim_t h(x+m(t) ,t)$ exists and is non-trivial. In particular, as in Bramson's results for non-linear equation, recover celebrated $-(3/2)\log t$ correction conditions decaying faster than $x^\nu e^{-x}$ some $\nu<-2$. Furthermore, when are this...

Motivated by the study of branching particle systems with selection, we establish global existence for solution free boundary problem when initial condition is non-increasing as and . We construct limit a sequence , where each un Fisher–KPP equation same condition, but different nonlinear term. Recent results De Masi A et al (2017 (arXiv:1707.00799)) show that this can be identified hydrodynamic so-called N-BBM, i.e. Brownian motion in which population size kept constant equal to N removing...

Le présent article a pour objet l'étude du processus extrémal mouvement Brownien branchant conditionné à avoir une particule anormalement loin droite. Ces mesures ponctuelles limites forment famille un paramètre et apparaissent dans les extrémaux de plusieurs branchement tels que le avec vitesse variable ou certains branchants multitype. Nous donnons nouvelle représentation ces nous montrons qu'elles continue lois. obtenons ainsi expression probabiliste simple la constante qui apparaît...

It has been conjectured since the work of Lalley and Sellke (1987) that branching Brownian motion seen from its tip (e.g. rightmost particle) converges to an invariant point process. Very recently, it emerged this can be proved in several different ways (see e.g. Brunet Derrida, 2010, Arguin et al., 2011). The structure extremal process turns out a Poisson with exponential intensity which each atom decorated by independent copy auxiliary main goal present is give complete description limit...

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

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