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For hypocoercive linear kinetic equations we first formulate an optimisationproblem on a spatially dependent jump rate in order to find the fastest decayrate of perturbations. In the Goldstein-Taylor model we show (i) that for alocally optimal jump rate the spectral gap is determined by multiple, possibledegenerate, eigenvectors and (ii) that globally the fastest decay is obtainedwith a spatially homogeneous jump rate. Our proofs rely on a connection todamped wave equations and a relationship to the spectral theory ofSchr{\"o}dinger operators.

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