Trajectorial dissipation of $\mathrm{\Phi }$-entropies for interacting particle systems

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 $\mathrm{\Phi }$-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. In this paper,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 ${\mathbb{R}}^n$ to an infinite-dimensionalsetting with weak assumptions on the dynamics.