Kingman's coalescent and Brownian motion

We describe a simple construction of Kingman's coalescent in terms of a Brownian excursion. This construction is closely related to, and sheds some new light on, earlier work by Aldous and Warren. Our approach also yields some new results: for instance, we obtain the full multifractal spectrum of Kingman's coalescent. This complements earlier work on Beta-coalescents by the authors and Schweinsberg. Surprisingly, the thick part of the spectrum is not obtained by taking the limit as $\alpha \to 2$ in the result for Beta-coalescents mentioned above. Other analogies and differences between the case of Beta-coalescents and Kingman's coalescent are discussed

KPP reaction-diffusion equations with a non-linear loss inside a cylinder

We consider in this paper a reaction-diffusion system in presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a non-linear spacedependent loss term inside the domain. In particular, we will point out the existence of a minimal speed, above which any real value is an admissible speed. We will also give some spreading results for initial conditions decaying exponentially at infinity

Traveling waves and homogeneous fragmentation

We formulate the notion of the classical Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu [In Seminar on Stochastic Processes (1988) 223-242 Birkh\"{a}user] and Chauvin [Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump-Mode-Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365-395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553-570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincar\'{e} Probab. Statist. 46 (2010) 119-134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609-631]) showing their mathematical robustness even within the context of fragmentation theory.Comment: Published in at http://dx.doi.org/10.1214/10-AAP733 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The branching Brownian motion seen from its tip

It has been conjectured since the work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, 2010, Arguin et al., 2010, 2011). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (2011).Comment: 47 pages, 3 figure

Survival of near-critical branching Brownian motion

Consider a system of particles performing branching Brownian motion with negative drift $\mu = \sqrt{2 - \epsilon}$ 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 $\epsilon>0$. Here we are interested in the asymptotics as \eps\to 0 of the survival probability $Q_\mu(x)$. It is proved that if $L= \pi/\sqrt{\epsilon}$ then for all $x \in \R$, $\lim_{\epsilon \to 0} Q_\mu(L+x) = \theta(x) \in (0,1)$ 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

Solitary wave dynamics in time-dependent potentials

We rigorously study the long time dynamics of solitary wave solutions of the nonlinear Schr\"odinger equation in {\it time-dependent} external potentials. To set the stage, we first establish the well-posedness of the Cauchy problem for a generalized nonautonomous nonlinear Schr\"odinger equation. We then show that in the {\it space-adiabatic} regime where the external potential varies slowly in space compared to the size of the soliton, the dynamics of the center of the soliton is described by Hamilton's equations, plus terms due to radiation damping. We finally remark on two physical applications of our analysis. The first is adiabatic transportation of solitons, and the second is Mathieu instability of trapped solitons due to time-periodic perturbations.Comment: 38 pages, some typos corrected, one reference added, one remark adde

Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection

In this work we continue the description of soliton-like solutions of some slowly varying, subcritical gKdV equations. In this opportunity we describe, almost completely, the allowed behaviors: either the soliton is refracted, or it is reflected by the potential, depending on its initial energy. This last result describes a new type of soliton-like solution for gKdV equations, also present in the NLS case. Moreover, we prove that the solution is not pure at infinity, unlike the standard gKdV soliton.Comment: 51 pages, submitte

The free energy in a class of quantum spin systems and interchange processes

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin $S=\tfrac12$ the model is the Heisenberg ferromagnet, for general spin $S\in\tfrac12\mathbb{N}$ it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by T\'oth and by Penrose when $S=\tfrac12$). The critical temperature is shown to coincide (as a function of $S$) with that of the $q=2S+1$ state classical Potts model, and the phase transition is discontinuous when $S\geq1$.Comment: 22 page

Existence and Bifurcation of Solutions for an Elliptic Degenerate Problem

AbstractWe investigate the existence, multiplicity and bifurcation of solutions of a model nonlinear degenerate elliptic differential equation: −x2u″=λu+|u|p−1uin (0, 1);u(0)=u(1)=0. This model is related to a simplified version of the nonlinear Wheeler–DeWitt equation as it appears in quantum cosmological models. We prove the existence of multiple positive solutions. More precisely, we show that there exists an infinite number of connected branches of solutions which bifurcate from the bottom of the essential spectrum of the corresponding linear operator.Nous étudions ici l'existence, multiplicité et propriétés de bifurcation des solutions d'un problème elliptique dégénéré: −x2u″=λu+|u|p−1uin (0, 1);u(0)=u(1)=0. Ce problème modèle est proche d'une version simplifiée et non-linéaire de l'équation de Wheeler–DeWitt, utilisée dans des modèles de Cosmologie quantique. Nous prouvons l'existence d'une infinité de branches de solutions qui bifurquent à partir de l'infimum du spectre continu de l'opérateur linéaire correspondant