Showing papers by "Nicolas Fournier published in 2013"

On the rate of convergence in Wasserstein distance of the empirical measure

Abstract: Let $\mu_N$ be the empirical measure associated to a $N$-sample of a given probability distribution $\mu$ on $\mathbb{R}^d$. We are interested in the rate of convergence of $\mu_N$ to $\mu$, when measured in the Wasserstein distance of order $p>0$. We provide some satisfying non-asymptotic $L^p$-bounds and concentration inequalities, for any values of $p>0$ and $d\geq 1$. We extend also the non asymptotic $L^p$-bounds to stationary $\rho$-mixing sequences, Markov chains, and to some interacting particle systems.

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Abstract: Nous etudions une equation differentielle stochastique de dimension $1$ dirigee par un processus de Levy stable. Lorsque $\alpha\in(1,2)$, nous examinons l’unicite trajectorielle pour cette equation. Quand $\alpha\in(0,1)$, nous etudions une autre equation, equivalente en loi, mais pour laquelle l’unicite trajectorielle s’avere vraie sous des hypotheses bien plus faibles. Nous obtenons des resultats varies, selon que $\alpha\in(0,1)$ ou $\alpha\in(1,2)$ et selon que le processus stable dirigeant l’equation est symetrique ou non. Nos hypotheses concernent la regularite et la monotonie des coefficients de derive et de diffusion.

Rate of convergence of the Nanbu particle system for hard potentials

Abstract: We consider the (numerically motivated) Nanbu stochastic particle system associated to the spatially homogeneous Boltzmann equation for true hard potentials. We establish a rate of propagation of chaos of the particle system to the unique solution of the Boltzmann equation. More precisely, we estimate the expectation of the squared Wasserstein distance with quadratic cost between the empirical measure of the particle system and the solution. The rate we obtain is almost optimal as a function of the number of particles but is not uniform in time.

One-Dimensional General Forest Fire Processes

Abstract: We consider the one-dimensional generalized forest fire process: at each site of $\mathbb{Z}$, seeds and matches fall according some i.i.d. stationary renewal processes. When a seed falls on an empty site, a tree grows immediately. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding connected component of occupied sites. Under some quite reasonable assumptions on the renewal processes, we show that when matches become less and less frequent, the process converges, with a correct normalization, to a limit forest fire model. According to the nature of the renewal processes governing seeds, there are four possible limit forest fire models. The four limit processes can be perfectly simulated. This study generalizes consequently a previous result of the authors where seeds and matches were assumed to fall according to Poisson processes.