Quantitative Rates of Convergence to Equilibrium for the Degenerate Linear Boltzmann equation on the Torus

TL;DR

This paper establishes quantitative convergence rates to equilibrium for a degenerate linear Boltzmann equation on the torus, under geometric control conditions, using probabilistic methods based on Markov chain theory.

Contribution

It provides explicit convergence rates for the equation under geometric control, extending previous qualitative results with a probabilistic approach.

Findings

Exponential convergence rates are obtained when the geometric control condition is met.

The approach uses Doeblin's theorem from Markov chain theory to quantify convergence.

Results apply to equations with spatially varying jump rates satisfying certain geometric conditions.

Abstract

We study the linear relaxation Boltzmann equation on the torus with a spatially varying jump rate which can be zero on large sections of the domain. In \cite{BS13} Bernard and Salvarani showed that this equation converges exponentially fast to equilibrium if and only if the jump rate satisfies the geometric control condition of Bardos, Lebeau and Rauch \cite{BLR91}. In \cite{HL15} Han-Kwan and L\'{e}autaud showed a more general result for linear Boltzmann equations under the action of potentials in different geometric contexts, including the case of unbounded velocities. In this paper we obtain quantitative rates of convergence to equilibrium when the geometric control condition is satisfied, using a probabilistic approach based on Doeblin's theorem from Markov chains.