Solving Landau equation for some soft potentials through a probabilistic approach

Abstract

This article deals with a way to solve the spatially homogeneous Landau equation using probabilistic tools. Thanks to the study of a nonlinear stochastic differential equation driven by a space-time white noise, we state the existence of a measure solution of the Landau equation with probability measure initial data, for a generalization of the Maxwellian molecules case. Then, by approximation of the Landau coefficients, the first result helps us to state the existence of a measure solution for some soft potentials [$\gamma \in ( -1,0) $]. This second part is based on the use of nonlinear stochastic differential equations and some martingale problems.