  We study hypocoercivity for a class of linearized BGK models for continuous
phase spaces. We develop methods for constructing entropy functionals that
enable us to prove exponential relaxation to equilibrium with explicit and
physically meaningful rates. In fact, we not only estimate the exponential
rate, but also the second time scale governing the time one must wait before
one begins to see the exponential relaxation in the L1 distance. This waiting
time phenomenon, with a long plateau before the exponential decay "kicks in"
when starting from initial data that is well-concentrated in phase space, is
familiar from work of Aldous and Diaconis on Markov chains, but is new in our
continuous phase space setting. Our strategies are based on the entropy and
spectral methods, and we introduce a new "index of hypocoercivity" that is
relevant to models of our type involving jump processes and not only diffusion.
At the heart of our method is a decomposition technique that allows us to adapt
Lyapunov's direct method to our continuous phase space setting in order to
construct our entropy functionals. These are used to obtain precise information
on linearized BGK models. Finally, we also prove local asymptotic stability of
a nonlinear BGK model.
