On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions

TL;DR

This paper establishes the global existence of renormalized martingale solutions for the Boltzmann equation with stochastic kinetic transport, under specific assumptions, advancing the mathematical understanding of stochastic kinetic models.

Contribution

It proves the existence of solutions to a stochastic Boltzmann equation with large initial data, using renormalization techniques and weak solution analysis.

Findings

Proved global existence of solutions under cut-off and coloring assumptions.

Developed a criterion for renormalization in stochastic kinetic equations.

Established tightness of velocity averages in $L^1$ space.

Abstract

This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in $L^1$.