The Landau Equation for Maxwellian molecules and the Brownian Motion on SO_R(N)

TL;DR

This paper provides a novel probabilistic representation of the Landau equation for Maxwellian molecules using Brownian motion on SO(N) and Gaussian processes, leading to precise bounds on transition densities.

Contribution

It introduces a new stochastic process representation for the Landau equation, connecting it with Brownian motion on rotation groups and Gaussian processes.

Findings

Established sharp multi-scale bounds for transition densities.

Linked the Landau equation to Brownian motion on SO(N).

Revealed the multi-scale structure depending on initial condition support.

Abstract

In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition.