Showing papers by "Nicolas Fournier published in 2010"
TL;DR: In this paper, an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes is introduced, which is based on a comparison between the Fourier transform of such marginals with those of the one-step Euler approximation of the underlying process.
Abstract: We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Holder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some Levy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable.
70 citations
TL;DR: In this article, the uniqueness of bounded solutions for the spatially homogeneous Fokker-Planck-Landau equation with a Coulomb potential was proved for the first time.
Abstract: We prove the uniqueness of bounded solutions for the spatially homogeneous Fokker-Planck-Landau equation with a Coulomb potential. Since the local (in time) existence of such solutions has been proved by Arsen’ev–Peskov (Z. Vycisl. Mat. i Mat. Fiz. 17:1063–1068, 1977), we deduce a local well-posedness result. The stability with respect to the initial condition is also checked.
47 citations
TL;DR: In this paper, the singularity spectrum of a pure jump Markov process is determined, which turns out to be random and to depend locally on the values taken by the process.
Abstract: We construct a nondecreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.
37 citations
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TL;DR: In this article, a one-dimensional stochastic differential equation driven by a stable Levy process of order with drift and diffusion coefficients was studied, and pathwise uniqueness was investigated.
Abstract: We study a one-dimensional stochastic differential equation driven by a stable Levy process of order $\alpha$ with drift and diffusion coefficients $b,\sigma$. When $\alpha\in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha\in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha\in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $\sigma$.
30 citations
TL;DR: In this article, the authors consider the one-dimensional forest fire process and show that when λ is made to tend to 0 with an appropriate normalization, the Forest Fire process tends to a uniquely defined process, the dynamics of which they precisely describe.
Abstract: We consider the so-called one-dimensional forest fire process. At each site of ℤ, a tree appears at rate 1. At each site of ℤ, a fire starts at rate λ>0, immediately destroying the whole corresponding connected component of trees. We show that when λ is made to tend to 0 with an appropriate normalization, the forest fire process tends to a uniquely defined process, the dynamics of which we precisely describe. The normalization consists of accelerating time by a factor log(1/λ) and of compressing space by a factor λ log(1/λ). The limit process is quite simple: it can be built using a graphical construction and can be perfectly simulated. Finally, we derive some asymptotic estimates (when λ→0) for the cluster-size distribution of the forest fire process.
13 citations
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TL;DR: In this article, the authors derived a satisfying rate of convergence of the Marcus-Lushnikov process toward the solution to Smoluchowski's coagulation equation for a class of homogeneous-like kernels with homogeneity degree ranging in $(-\infty,1]$.
Abstract: We derive a satisfying rate of convergence of the Marcus-Lushnikov process toward the solution to Smoluchowski's coagulation equation. Our result applies to a class of homogeneous-like coagulation kernels with homogeneity degree ranging in $(-\infty,1]$. It relies on the use of a Wasserstein-type distance, which has shown to be particularly well-adapted to coalescence phenomena.
10 citations
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TL;DR: In this paper, the authors considered the white-noise driven stochastic heat equation with Lipschitz-continuous drift and diffusion coefficients and derived an inequality for the difference between two solutions.
Abstract: We consider the white-noise driven stochastic heat equation on $[0,\infty)\times[0,1]$ with Lipschitz-continuous drift and diffusion coefficients $b$ and $\sigma$. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some {\it a priori} estimates on solutions. This allows us to prove the strong existence and (partial) uniqueness of weak solutions when the initial condition belongs only to $L^1([0,1])$, and the stability of the solution with respect to this initial condition. We also obtain, under some conditions, some results concerning the large time behavior of solutions: uniqueness of the possible invariant distribution and asymptotic confluence of solutions.
2 citations