Showing papers by "Nicolas Fournier published in 2000"
TL;DR: In this article, the authors consider a two-dimensional Kac equation without cutoff, which they relate to a stochastic differential equation, and prove the existence of a solution for this SDE, and use the Malliavin calculus to prove that the law of this solution admits a smooth density with respect to the Lebesgue measure.
Abstract: We consider a two-dimensional Kac equation without cutoff,which we relate to a stochastic differential equation.We prove the existence of a solution for this SDE, and we use the Malliavin calculus (or stochastic calculus of variations) to prove that the law of this solution admits a smooth density with respect to the Lebesgue measure on $\mathbf{R}^2$.This density satisfies the Kac equation.
30 citations
TL;DR: In this paper, the authors studied a parabolic SPDE driven by a white noise and a compensated Poisson measure, and proved the existence and uniqueness of a weak solution in the Malliavin calculus.
29 citations
TL;DR: In this paper, the authors considered the Malliavin calculus for Poisson functionals and proved that this solution is strictly positive on ]0,∞[xℝ.
Abstract: We consider the solution of a one-dimensional Kac equation without cutoff built by Graham and Meleard. Recalling that this solution is the density of a Poisson driven nonlinear stochastic differential equation, we develop Bismut's approach of the Malliavin calculus for Poisson functionals in order to prove that this solution is strictly positive on ]0,∞[xℝ.
8 citations