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)

A small-time coupling between Λ\Lambda-coalescents and branching processes

We describe a new general connection between $\Lambda$-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 $N^{\Lambda}(t)$ to the classical upper and lower indices arising in the study of pathwise properties of L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/12-AAP911 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP728 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Propagation phenomena for time heterogeneous KPP reaction-diffusion equations

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation $\partial_t u -\Delta u = f(t,u)$, $x\in R^N$, $t\in\R$, where f=f(t,u) is a KPP monostable nonlinearity which depends in a general way on t. A typical f which satisfies our hypotheses is f(t,u)=m(t) u(1-u), with m bounded and having positive infimum. We first prove the existence of generalized transition waves (recently defined by Berestycki and Hamel, Shen) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t. Lastly, we prove some spreading properties for the solution of the Cauchy problem

Beta-coalescents and continuous stable random trees

Coalescents with multiple collisions, also known as $\Lambda$-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure $\Lambda$ is the $\operatorname {Beta}(2-\alpha,\alpha)$ distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly--Kurtz lookdown process using continuous random trees, which is of independent interest. This produces a number of results concerning the small-time behavior of Beta-coalescents. Most notably, we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics.Comment: Published in at http://dx.doi.org/10.1214/009117906000001114 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Small-time behavior of beta coalescents

For a finite measure $\varLambda$ on $[0,1]$, the $\varLambda$-coalescent is a 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 $1<\alpha<2$, the $\varLambda$-coalescent in which $\varLambda$ is the $\operatorname {Beta}(2-\alpha,\alpha)$ distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an $\alpha$-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other $\varLambda$-coalescents for which $\varLambda$ has the same asymptotic behavior near zero as the $\operatorname {Beta}(2-\alpha,\alpha)$ distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of $\varLambda$-coalescents.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP103 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

The hyperbolic geometry of random transpositions

Turn the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\sigma_t$ be the simple random walk on this graph. In a previous paper, Berestycki and Durrett [In Discrete Random Walks (2005) 17--26] showed that the limiting behavior of the distance from the identity at time $cn/2$ has a phase transition at $c=1$. Here we investigate some consequences of this result for the geometry of $G_n$. Our first result can be interpreted as a breakdown for the Gromov hyperbolicity of the graph as seen by the random walk, which occurs at a critical radius equal to $n/4$. Let $T$ be a triangle formed by the origin and two points sampled independently from the hitting distribution on the sphere of radius $an$ for a constant $0<a<1$. Then when $a<1/4$, if the geodesics are suitably chosen, with high probability $T$ is $\delta$-thin for some $\delta>0$, whereas it is always O(n)-thick when $a>1/4$. We also show that the hitting distribution of the sphere of radius $an$ is asymptotically singular with respect to the uniform distribution. Finally, we prove that the critical behavior of this Gromov-like hyperbolicity constant persists if the two endpoints are sampled from the uniform measure on the sphere of radius $an$. However, in this case, the critical radius is $a=1-\log2$.Comment: Published at http://dx.doi.org/10.1214/009117906000000043 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org