Finding the jump rate for fastest decay in the Goldstein-Taylor model

TL;DR

This paper formulates an optimization problem to identify the spatially dependent jump rate that maximizes decay in hypocoercive kinetic equations, revealing that the fastest decay occurs with a uniform jump rate in the Goldstein-Taylor model.

Contribution

It introduces a novel optimization framework for jump rates in kinetic equations and demonstrates that the globally optimal decay rate is achieved with a spatially homogeneous jump rate.

Findings

Spectral gap determined by multiple eigenvectors for locally optimal jump rates.

Globally fastest decay achieved with spatially uniform jump rate.

Connection established between kinetic decay and spectral theory of Schrödinger operators.

Abstract

For hypocoercive linear kinetic equations we first formulate an optimisation problem on a spatially dependent jump rate in order to find the fastest decay rate of perturbations. In the Goldstein-Taylor model we show (i) that for a locally optimal jump rate the spectral gap is determined by multiple, possible degenerate, eigenvectors and (ii) that globally the fastest decay is obtained with a spatially homogeneous jump rate. Our proofs rely on a connection to damped wave equations and a relationship to the spectral theory of Schr{\"o}dinger operators.