Non-reversible processes: GENERIC, Hypocoercivity and fluctuations

TL;DR

This paper compares Hypocoercivity Theory and GENERIC approaches for analyzing non-reversible Markov processes, providing explicit formulas, linking reversibility with gradient flow structures, and applying results to diffusion and PDMPs.

Contribution

It offers a unified comparison of HT and GENERIC, explicit formulas for transition, and new results on entropy decay for PDMPs.

Findings

Explicit formulas linking HT and GENERIC

Proof of reversibility-gradient flow connection

Entropy decay in certain PDMPs

Abstract

We consider two approaches to study non-reversible Markov processes, namely the Hypocoercivity Theory (HT) and GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling); the basic idea behind both of them is to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker-Planck equation. We do this both for linear Markov processes and for a class of nonlinear Markov process as well. We then characterize the structure of the Large deviation functional of generalised-reversible processes; this is a class of non-reversible processes of large relevance in applications. Finally, we show how our results apply to two classes of Markov processes, namely non-reversible diffusion processes and a class of Piecewise Deterministic Markov Processes (PDMPs), which have recently attracted the attention of the statistical sampling community. In particular, for the PDMPs we consider we prove entropy decay.