arXiv subjects

Jean Bertoin

Publications and source records attributed to Jean Bertoin.

At least 37 records · Page 2Linked to original sources

arXiv2019

On conditioning a self-similar growth-fragmentation by its intrinsic area

The genealogical structure of self-similar growth-fragmentations can be described in terms of a branching random walk. The so-called intrinsic area $\mathrm{A}$ arises in this setting as the terminal value of a remarkable additive martingale. Motivated by connections with some models of random planar geometry, the purpose of this work is to investigate the effect of conditioning a self-similar growth-fragmentation on its intrinsic area. The distribution of $\mathrm{A}$ satisfies a useful smoothing transform which enables us to establish the existence of a regular density $a$ and to determine the asymptotic behavior of $a(r)$ as $r\to \infty$ (this can be seen as a local version of Kesten-Grincevicius-Goldie theorem's for random affine fixed point equations in a particular setting). In turn, this yields a family of martingales from which the formal conditioning on $\mathrm{A}=r$ can be realized by probability tilting. We point at a limit theorem for the conditional distribution given $\mathrm{A}=r$ as $r\to \infty$, and also observe that such conditioning still makes sense under the so-called canonical measure for which the growth-fragmentation starts from $0$

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arXiv2019

A version of Herbert A. Simon's model with slowly fading memory and its connections to branching processes

Construct recursively a long string of words w1. .. wn, such that at each step k, w k+1 is a new word with a fixed probability p $\in$ (0, 1), and repeats some preceding word with complementary probability 1 -- p. More precisely, given a repetition occurs, w k+1 repeats the j-th word with probability proportional to j $α$ for j = 1,. .. , k. We show that the proportion of distinct words occurring exactly times converges as the length n of the string goes to infinity to some probability mass function in the variable $\ge$ 1, whose tail decays as a power function when 1 -- p > $α$/(1 + $α$), and exponentially fast when 1 -- p < $α$/(1 + $α$).

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arXiv2019

The strong Malthusian behavior of growth-fragmentation processes

Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. Typically, we may expect that in the long run, the concentrations of cells with given masses increase at some exponential rate, and that, after compensating for this, they arrive at an asymptotic profile. Up to now, this question has mainly been studied for the average behavior of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behavior of the system as a whole, rather than only its average, is more delicate. In this work, we show that a criterion found by one of the authors for exponential ergodicity on average is actually sufficient to deduce stronger results about the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur.

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arXiv2018

Noise reinforcement for L{é}vy processes

In a step reinforced random walk, at each integer time and with a fixed probability p $\in$ (0, 1), the walker repeats one of his previous steps chosen uniformly at random, and with complementary probability 1 -- p, the walker makes an independent new step with a given distribution. Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a L{é}vy process. For sub-critical (or admissible) memory parameters p < p c , where p c is related to the Blumenthal-Getoor index of the L{é}vy process, we construct a noise reinforced L{é}vy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the L{é}vy process, converge weakly to the noise reinforced L{é}vy process as the time-mesh goes to 0.

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arXiv2018

Infinitely ramified point measures and branching Lévy processes

We call a random point measure infinitely ramified if for every $n\in \mathbb N$, it has the same distribution as the $n$-th generation of some branching random walk. On the other hand, branching Lévy processes model the evolution of a population in continuous time, such that individuals move in space independently, according to some Lévy process, and further beget progenies according to some Poissonian dynamics, possibly on an everywhere dense set of times. Our main result connects these two classes of processes much in the same way as in the case of infinitely divisible distributions and Lévy processes: the value at time $1$ of a branching Lévy process is an infinitely ramified point measure, and conversely, any infinitely ramified point measure can be obtained as the value at time $1$ of some branching Lévy process.

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arXiv2018

Biggins' Martingale Convergence for Branching Lévy Processes

A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $(σ^2,a,Λ)$, where the branching Lévy measure $Λ$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(σ^2,a,Λ)$ for additive martingales to have a non-degenerate limit.

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arXiv2018

Branching-stable point measures and processes

We introduce and study the class of branching-stable point measures, which can be seen as an analog of stable random variables when the branching mechanism for point measures replaces the usual addition. In contrast with the classical theory of stable (Lévy) processes, there exists a rich family of branching-stable point measures with \emph{negative} scaling exponent, which can be described as certain Crump-Mode-Jagers branching processes. We investigate the asymptotic behavior of their cumulative distribution functions, that is, the number of atoms in $(-\infty, x]$ as $x\to \infty$, and further depict the genealogical lineage of typical atoms. For both results, we rely crucially on the work of Biggins.

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arXiv2018

On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors

This work develops further a probabilist approach to the asymptotic behavior of growth-fragmentation semigroups via the Feynman-Kac formula, which was introduced in a joint article with A.R. Watson [4]. Here, it is first shown that the sufficient condition for a Malthusian behavior which was established in [4], is also necessary. We then provide a simple criterion to ensure exponential speed of convergence, which enables us to treat cases than were not covered previously in the literature.

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arXiv2017

Martingales in self-similar growth-fragmentations and their connections with random planar maps

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall & Miermont. As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in arXiv:1507.02265 .

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arXiv2017

Ergodic aspects of some Ornstein-Uhlenbeck type processes related to Lévy processes

This work concerns the Ornstein-Uhlenbeck type process associated to a positive self-similar Markov process $(X(t))_{t\geq 0}$ which drifts to $\infty$, namely $U(t):= {\rm e}^{-t}X({\rm e}^t-1)$. We point out that $U$ is always a (topologically) recurrent Markov process and identify its invariant measure in terms of the law of the exponential functional $\hat I := \int_0^\infty \exp(\hatξ_s) {\rm d}s$, where $\hatξ$ is the dual of the real-valued Lévy process $ξ$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $ξ_1\not \in L^1(\mathbb{P})$. In that case, we determine the family of Lévy processes $ξ$ for which $U$ fulfills the conclusions of the Darling-Kac theorem. Our approach relies crucially on another generalized Ornstein-Uhlenbeck process that can be associated to the Lévy process $ξ$, namely $V(t) := \exp(ξ_t)\left(\int_0^t \exp(-ξ_s){\rm d}s +V(0)\right)$, and properties of time-substitutions based on additive functionals.

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arXiv2017

Random planar maps & growth-fragmentations

We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.

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arXiv2017

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymp-totic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach to the study of this asymptotic behaviour. We use a Feynman--Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the spectral radius and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. In special cases, we obtain exponential convergence.

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arXiv2016

Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

We consider a Yule process until the total population reaches size $n\gg 1$, and assume that neutral mutations occur with high probability $1-p$ (in the sense that each child is a new mutant with probability $1-p$, independently of the other children), where $p=p_n\ll 1$. We establish a general strategy for obtaining Poisson limit laws for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter $p_n$ tending to zero.

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arXiv2016

Local explosion in self-similar growth-fragmentation processes

Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in a work of Bertoin, where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.

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arXiv2015

Self-similar scaling limits of Markov chains on the positive integers

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits different behavior. The present results extend those of Haas & Miermont who focused on the case of non-increasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive, and exchangeable coalescence-fragmentation processes.

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arXiv2015

Probabilistic aspects of critical growth-fragmentation equations

The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered by Doumic and Escobedo in the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the non-homogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter $(0,\infty)$ continuously from either $0$ or $\infty$, we exhibit unexpected spontaneous generation of mass in the solutions.

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arXiv2015

The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes

We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time t is encoded by a partition $Π$(t) of N into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $Π$ is Markovian with transitions determined by a splitting rates measure r. However, somewhat surprisingly, r fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of $Π$(t). We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.

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arXiv2014

Cutting edges at random in large recursive trees

We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT's and present recent results on the cluster sizes in the supercritical regime.

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