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

This paper is about the rate of convergence to equilibrium for hypocoercive linear kinetic equations. We look for the spatially dependent jump rate which yields the fastest decay rate of perturbations. For the Goldstein–Taylor model, we show (i) that for a locally optimal jump rate the spectral bound is determined by multiple, possibly 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 Schrödinger operators