Trajectorial dissipation of $Phi$-entropies for interacting particle systems

Abstract

A classical approach for the analysis of the longtime behavior of Markov processes is to consider suitable Lyapunov functionals like the variance or more generally Φ -entropies. Via purely analytic arguments it can be shown that these functionals are indeed non-increasing in time under quite general assumptions on the process. We complement these classical results via a more probabilistic approach and show that dissipation is already present on the level of individual trajectories for spatially-extended systems of infinitely many interacting particles with arbitrary underlying geometry and compact local spin spaces. This extends previous results from the setting of finite-state Markov chains or diffusions in Rn to an infinite-dimensional setting with weak assumptions on the dynamics.