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Jean Bertoin

Publications and source records attributed to Jean Bertoin.

At least 73 records · Page 4Linked to original sources
arXiv2009

A system of grabbing particles related to Galton-Watson trees

We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree.

math.PR↗
arXiv2008

Two solvable systems of coagulation equations with limited aggregations

We consider two simple models for the formation of polymers where at the initial time, each monomer has a certain number of potential links (called arms in the text) that are consumed when aggregations occur. Loosely speaking, this imposes restrictions on the number of aggregations. The dynamics of concentrations are governed by modifications of Smoluchowski's coagulation equations. Applying classical techniques based on generating functions, resolution of quasi-linear PDE's, and Lagrange inversion formula, we obtain explicit solutions to these non-linear systems of ODE's. We also discuss the asymptotic behavior of the solutions and point at some connexions with certain known solutions to Smoluchowski's coagulation equations with additive or multiplicative kernels.

math-ph↗
arXiv2008

The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.

math.PR↗
arXiv2008

Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterise when a Lévy process $X_t$ crosses boundaries like $t^κ$, $κ>0$, in a one or two-sided sense, for small times $t$; thus, we enquire when $\limsup_{t\downarrow 0}|X_t|/t^κ$, $\limsup_{t\downarrow 0}X_t/t^κ$ and/or $\liminf_{t\downarrow 0}X_t/t^κ$ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of $κ>0$. Often (for many values of $κ$), when the limsups are finite a.s., they are in fact zero, as we show, but the limsups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one or two-sided boundaries in quite different ways, but surprisingly this is not so for the case $κ=1/2$. An integral test is given to distinguish the possibilities in that case. Some results relating to other norming sequences for $X$, and when $X$ is centered at a nonstochastic function, are also given.

math.PR↗
arXiv2007

Asymptotic regimes for the occupancy scheme of multiplicative cascades

In the classical occupancy scheme, one considers a fixed discrete probability measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls independently at random in boxes labeled by ${\cal I}$, such that $p_i$ is the probability that a given ball falls into the box $i$. In this work, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence $({\bf p}(k) : k\in\N)$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chains

math.PR↗
arXiv2006

Homogenenous Multitype Fragmentations

A homogeneous mass-fragmentation, as it has been defined in \cite{RFC}, describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the characteristics of the fragmentation. As applications, we make explicit the connexion with multitype branching random walks, and obtain multitype analogs of the pathwise central limit theorem and large deviation estimates for the empirical distribution of fragments.

math.PR↗
arXiv2006

On Continuity Properties of the Law of Integrals of Lévy Processes

Let $(ξ,η)$ be a bivariate Lévy process such that the integral $\int\_0^\infty e^{-ξ\_{t-}} dη\_t$ converges almost surely. We characterise, in terms of their \LL measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form $I:=\int\_0^\infty g(ξ\_t) dt$, where $g$ is a deterministic function. We give sufficient conditions ensuring that $I$ has no atoms, and under further conditions derive that $I$ has a Lebesgue density. The results are also extended to certain integrals of the form $\int\_0^\infty g(ξ\_t) dY\_t$, where $Y$ is an almost surely strictly increasing stochastic process, independent of $ξ$.

math.PR↗
arXiv2006

Reflecting a Langevin Process at an Absorbing Boundary

We consider a Langevin process with white noise random forcing. We suppose that the energy of the particle is instantaneously absorbed when it hits some fixed obstacle. We show that nonetheless, the particle can be instantaneously reflected, and study some properties of this reflecting solution.

math.PR↗
arXiv2005

Stochastic flows associated to coalescent processes III: Limit theorems

We prove several limit theorems that relate coalescent processes to continuous-state branching processes. Some of these theorems are stated in terms of the so-called generalized Fleming-Viot processes, which describe the evolution of a population with fixed size, and are duals to the coalescents with multiple collisions studied by Pitman and others. We first discuss asymptotics when the initial size of the population tends to infinity. In that setting, under appropriate hypotheses, we show that a rescaled version of the generalized Fleming-Viot process converges weakly to a continuous-state branching process. As a corollary, we get a hydrodynamic limit for certain sequences of coalescents with multiple collisions: Under an appropriate scaling, the empirical measure associated with sizes of the blocks converges to a (deterministic) limit which solves a generalized form of Smoluchowski's coagulation equation. We also study the behavior in small time of a fixed coalescent with multiple collisions, under a regular variation assumption on the tail of the measure $ν$ governing the coalescence events. Precisely, we prove that the number of blocks with size less than $εx$ at time $(εν([ε,1]))^{-1}$ behaves like $ε^{-1}λ\_1(]0,x[)$ as $ε\to 0$, where $λ\_1$ is the distribution of the size of one cluster at time 1 in a continuous-state branching process with stable branching mechanism. This generalizes a classical result for the Kingman coalescent.

math.PR↗
arXiv2005

Asymptotics in Knuth's parking problem for caravans

We consider a generalized version of Knuth's parking problem, in which caravans consisting of a number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravan's size tail distribution, several qualitatively different versions of eternal additive coalescent are involved.

math.PR↗
arXiv2005

Different Aspects of a Model for Random Fragmentation Processes

This text surveys different probabilistic aspects of a model which is used to describe the evolution of an object that falls apart randomly as time passes. Each point of view yields useful techniques to establish properties of such random fragmentation processes.

math.PR↗
arXiv2004

Asymptotical behaviour of the presence probability in branching random walks and fragmentations

For a subcritical Galton-Watson process $(ζ_n)$, it is well known that under an $X \log X$ condition, the quotient $P(ζ_n > 0)/ Eζ_n$ has a finite positive limit. There is an analogous result for a (one-dimensional) supercritical branching random walk: when $a$ is in the so-called subcritical speed area, the probability of presence around $na$ in the $n$-th generation is asymptotically proportional to the corresponding expectation. In Rouault (1993) this result was stated under a natural $X \log X$ assumption on the offspring point process and a (unnatural) condition on the offspring mean. Here we prove that the result holds without this latter condition, in particular we allow an infinite mean and a dimension $d \geq 1$ for the state-space. As a consequence the result holds also for homogeneous fragmentations as defined in Bertoin (2001), using the method of discrete-time skeletons; this completes the proof of Theorem 4 in Bertoin-Rouault (2004 see math/PR/0409545). Finally, an application to conditioning on the presence allows to meet again the probability tilting and the so-called additive martingale.

math.PR↗
arXiv2004

Discretization methods for homogeneous fragmentations

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogs of a certain type of branching random walks, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations. On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\N$. Spatial discretization is especially well-suited to develop directly for continuous times the conceptual method of probability tilting of Lyons, Pemantle and Peres.

math.PR↗