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

TL;DR

This paper demonstrates that $\

Contribution

It introduces a probabilistic method showing trajectory-level dissipation of $\

Findings

Dissipation occurs at the individual trajectory level.

Extends results from finite-state systems to infinite-dimensional systems.

Applicable to systems with arbitrary geometry and compact local spaces.

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 $\Phi$-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.