Grandes déviations pour une famille de processus de Galton-Watson dépendant de l'effectif de la population
Hierarchical equilibria of branching populations.
Homogeneous diffusions on the Sierpinski gasket
Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst.
Intermittency and ageing for the symbiotic branching model
For the symbiotic branching model introduced in [Stochastic Process. Appl.114 (2004) 127–160], it is shown that ageing and intermittency exhibit different behaviour for negative, zero, and positive correlations. Our approach also provides an alternative, elementary proof and refinements of classical results concerning second moments of the parabolic Anderson model with brownian potential. Some refinements to more general (also infinite range) kernels of recent ageing results of [Ann. Inst. H. Poincaré...
Intertwining of birth-and-death processes
It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues,...
Invariance principles for spatial multitype Galton–Watson trees
We prove that critical multitype Galton–Watson trees converge after rescaling to the brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the...
Karamata's Iteration Theorem And Normed Regularly Varying Sequences In A Historical Light
La fonction de Green d'un processus de Galton-Watson
La σ-algèbre asymptotique d'une chaîne de Galton-Watson
LAN and LAMN for systems of interacting diffusions with branching and immigration
Large deviation probabilities for some rescaled superprocesses
Large deviations for transient random walks in random environment on a Galton–Watson tree
Consider a random walk in random environment on a supercritical Galton–Watson tree, and let τn be the hitting time of generation n. The paper presents a large deviation principle for τn/n, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time.
Large deviations of Markov chains indexed by random trees
Large population limit and time behaviour of a stochastic particle model describing an age-structured population
We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound...
Law of large numbers for a class of superdiffusions
Law of large numbers for superdiffusions : the non-ergodic case
In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of the Watanabe–Biggins LLN for super-brownian motion.
Lévy processes conditioned on having a large height process
In the present work, we consider spectrally positive Lévy processes not drifting to and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with ) before hitting . This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law of this conditioned process (starting at ) is defined as a Doob -transform via a martingale. For Lévy processes with infinite variation paths, this martingale...
Likely path to extinction in simple branching models with large initial population.
Limit distributions for multitype branching processes of -ary search trees
Let be an integer. The so-called-ary search treeis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when , the asymptotic behavior of the process is Gaussian, but for it is no longer Gaussian and a limit of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version...