Jump processes as Generalized Gradient Flows

TL;DR

This paper develops a new functional framework for non-metric gradient systems, including Markov jump processes, providing definitions, existence proofs, and uniqueness results without relying on metric structures.

Contribution

It introduces a novel approach to analyze jump processes as generalized gradient flows using only dissipation functionals, without metric assumptions.

Findings

Established a solution notion for non-metric gradient systems.

Proved existence of solutions within the framework.

Provided an archetype uniqueness result for the systems.

Abstract

We have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure.