Sylvie Méléard

Chaos hypothesis for a system interacting through shared resources

SummaryWe study the chaos hypothesis for a wide class of pure-jump multitype interacting systems. The interaction may be strong, there is no symmetry assumption, and the system is not necessarily Markovian. We use interaction graphs and coupling and study in a precise way how a chain reaction is constituted by a series of direct interactions. We obtain the chaos hypothesis in variation norm with speed of convergence and deduce from it convergence of general empirical measures. We couple the interaction graph to a Boltzmann tree and show that the variation norm between the processes constructed on each goes to zero. This proves propagation of chaos in total variation with speed of convergence when the Boltzmann trees converge. Under light symmetry assumptions, we characterize the limit law by a nonlinear martingale problem.

Propagation of chaos and fluctuations for a moderate model with smooth initial data

Abstract In this paper, we are interested in a stochastic differential equation which is nonlinear in the following sense: both the diffusion and the drift coefficients depend locally on the density of the time marginal of the solution. When the law of the initial data has a smooth density with respect to Lebesgue measure, we prove existence and uniqueness for this equation. Under more restrictive assumptions on the density, we approximate the solution by a system of n moderately interacting diffusion processes and obtain a trajectorial propagation of chaos result. Finally, we study the fluctuations associated with the convergence of the empirical measure of the system to the law of the solution of the nonlinear equation. In this situation, the convergence rate is different from √ n .

Probabilistic Interpretation and Numerical Approximation of a Kac Equation without Cutoff

A nonlinear pure-jump Markov process is associated with a singular Kac equation. This process is the unique solution in law for a nonclassical stochastic differential equation. Its law is approximated by simulable stochastic particle systems, with rates of convergence. An effective numerical study is given at the end of the paper.

Propagation of chaos for a fully connected loss network with alternate routing

We study a stochastic loss network of switched circuits with alternate routing. The processes of interest will be the loads of the links, forming a strongly interacting system which is neither exchangeable nor Markovian. We consider interaction graphs representing the past history of a collection of links and prove their convergence to a limit tree, using the notion of chain reactions. Thus we prove a propagation of chaos result in variation norm for the laws of the whole sample paths, for general initial conditions, and in the i.i.d. case we have speeds of convergence.

Individual-based probabilistic models of adaptive evolution and various scaling approximations

We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individuals trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. We look for tractable large population approximations. By combining various scalings on population size, birth and death rates, mutation rate, mutation step, or time, a single microscopic model is shown to lead to contrasting macroscopic limits, of different nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. In the limit of rare mutations, we show that a possible approximation is a jump process, justifying rigorously the so-called trait substitution sequence. We thus unify different points of view concerning mutation-selection evolutionary models.

Convergence of the fluctuations for interacting diffusions with jumps associated with boltzmann equations

We consider a nonlinear partial differential equation of the Boltzmann type, in which the nonlinearity appears in a bounded jump operator.We associate with this equation a probability measure P on the Skorohod path space, solution of a nonlinear martingale problem and with time-marginals (Pt ) solutions of the pde. Then some interacting stochastic particle systems are given, whose empirical measures converge to P In this paper we are interested in the convergence of the fluctuation processes We prove by martingale techniques and Sobolev embeddings that the processes converge in law to a generalized Ornstein-Uhlenbeck process in the space where W is a weighted Sobolev space completely described

A stochastic particle numerical method for 3D Boltzmann equations without cutoff

Using the main ideas of Tanaka, the measure-solution {Pt}t of a 3-dimensional spatially homogeneous Boltzmann equation of Maxwellian molecules without cutoff is related to a Poisson-driven stochastic differential equation. Using this tool, the convergence to {Pt}t of solutions {Ptl}t of approximating Boltzmann equations with cutoff is proved, Then, a result of Graham-Meleard is used and allows us to approximate {Ptl}t with the empirical measure {µtl,n}t of an easily simulable interacting particle system. Precise rates of convergence are given. A numerical study lies at the end of the paper.

A Hilbertian approach for fluctuations on the McKean-Vlasov model

We consider the sequence of fluctuation processes associated with the empirical measures of the interacting particle system approximating the d-dimensional McKean-Vlasov equation and prove that they are tight as continuous processes with values in a precise weighted Sobolev space. More precisely, we prove that these fluctuations belong uniformly (with respect to the size of the system and to time) to W-(1+D), 2D0 and converge in C([0, T], W-(2+2D), D0) to a Ornstein-Uhlenbeck process obtained as the solution of a Langevin equation in W-(4+2D), D0, where D is equal to 1 + [d/2]. It appears in the proofs that the spaces W-(1 --> D), 2D0 and W-(2-2D), D0 are minimal Sobolev spaces in which to immerse the fluctuations, which was our aim following a physical point of view.

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,(18) showed a way to relate the measure solution {Pt}t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {Pt} is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.

An Ergodic Result for Critical Spatial Branching Processes

The infinite particle system known as spatial branching process has been introduced (under the name Branching Markov process) by Ikeda, Nagasawa, Watanabe ([I-N-W]).