Trajectorial Dissipation of {Phi }-Entropies for Interacting Particle Systems

Abstract

A classical approach for the analysis of the long-time behaviour of Markov processes is to consider suitable Lyapunov functionals like the variance or more generally Phi -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 refine 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 mathbb {R}^n to an infinite-dimensional setting with weak assumptions on the dynamics.