Young Researchers Workshop: Stochastic and deterministic methods in kinetic theory

The Boltzmann equation with stochastic kinetic tranport

Scott Smith

Max Planck Institute for Mathematics in the Sciences


The study of stochastic fluid dynamics has seen many developments in recent years. A number of authors have considered the existence problem for Navier-Stokes and Euler equations with random perturbations. In this framework, the fluid is modelled by a stochastic partial differential equation (SPDE) describing the time evolution of the macroscopic fluid variables in the presence of an external noise. In this talk, we consider a related SPDE, but from the mesocopic viewpoint of the Boltzmann equation. Namely, we allow for a background, or environmental noise to act on particles in between collisions. At the level of the kinetic description, this leads to a very particular form of noise, referred to as stochastic kinetic transport. We will discuss recent results on the existence of renormalized (in the sense of Di-Perna/Lions) martingale (in the sense of Stroock/Varadhan) solutions to this SPDE. The approach is based on a detailed analysis of weak martingale solutions to linear stochastic tranport equations driven by a random source; including criteria for renormalization, velocity avearging, and weak compactness of the solution set. Time permitting, we will also discuss the formal connection between the mesocopic and macroscopic viewpoints, via the moments method of Bardos/Golse/Levermore for hydrodynamic limits. This is a joint work with Sam Punshon-Smith (University of Maryland).