Hypocoercivity of the linearized BGK-equation with stochastic coefficients

TL;DR

This paper investigates how randomness affects the decay to equilibrium in a linearized BGK model, demonstrating exponential decay rates that are robust to stochastic influences using Lyapunov's method.

Contribution

It establishes the exponential decay rate for the stochastic linearized BGK model and shows its independence from stochastic effects, extending hypocoercivity analysis to stochastic coefficients.

Findings

Exponential decay rate proven for the stochastic BGK model

Decay rate is independent of stochastic influence in a physical norm

Analysis of derivatives with respect to stochastic variables

Abstract

In this paper we study the effect of randomness on a linearized BGK-model in one dimension. We prove exponential decay rate to a global equilibrium. This decay rate can be proven to be independent of the stochastic influence in a physical reasonable norm. We will further discuss the decay rate of the $n$-th derivative with respect to the stochastic variable of the solutions. Our strategy is based on Lyapunov's method. The matrices we need for a Lyapunov's estimate now depend on the stochastic variable. This requires a careful analysis of the random effect.