Particle approximation of some Landau equations

TL;DR

This paper introduces a particle-based numerical scheme for solving certain Landau equations, providing a probabilistic interpretation and analyzing convergence, with extensions to soft potentials and numerical demonstrations.

Contribution

It presents a novel particle approximation method for Landau equations, extending probabilistic interpretations and analyzing convergence rates.

Findings

Proposed a particle system driven by Brownian motions for Landau equations.

Derived convergence rates for the numerical scheme.

Extended the method to soft potentials with numerical results.

Abstract

We consider a class of nonlinear partial-differential equations, including the spatially homogeneous Fokker-Planck-Landau equation for Maxwell (or pseudo-Maxwell) molecules. Continuing the work of Fontbona-Gu\'erin-M\'el\'eard, we propose a probabilistic interpretation of such a P.D.E. in terms of a nonlinear stochastic differential equation driven by a standard Brownian motion. We derive a numerical scheme, based on a system of $n$ particles driven by $n$ Brownian motions, and study its rate of convergence. We finally deal with the possible extension of our numerical scheme to the case of the Landau equation for soft potentials, and give some numerical results.