Characterisation of gradient flows on finite state Markov chains

TL;DR

This paper characterizes when the evolution of finite state Markov chains can be described as a gradient flow, establishing a connection with time-reversibility and the structure of the generator.

Contribution

It proves the converse of Maas's 2011 result, showing gradient flow evolution implies time-reversibility, and extends the characterization to general functionals and generator properties.

Findings

Gradient flow evolution implies time-reversibility.

Evolution as gradient flow with general functionals requires diagonalisable generator.

Discussion on functional aspects uniquely determined by the Markov chain.

Abstract

In his 2011 work, Maas has shown that the law of any time-reversible continuous-time Markov chain with finite state space evolves like a gradient flow of the relative entropy with respect to its stationary distribution. In this work we show the converse to the above by showing that if the relative law of a Markov chain with finite state space evolves like a gradient flow of the relative entropy functional, it must be time-reversible. When we allow general functionals in place of the relative entropy, we show that the law of a Markov chain evolves as gradient flow if and only if the generator of the Markov chain is real diagonalisable. Finally, we discuss what aspects of the functional are uniquely determined by the Markov chain.