Evolution of the Wasserstein distance between the marginals of two Markov processes

TL;DR

This paper derives a formula for the time derivative of the Wasserstein distance between marginals of Markov processes, demonstrating its validity for jump processes and certain deterministic processes, with applications to birth-death processes.

Contribution

It provides a rigorous proof for the evolution of Wasserstein distance in jump and deterministic Markov processes, extending previous theoretical understanding.

Findings

The Wasserstein distance evolution is given by a specific integral formula.

The formula holds for pure jump processes with bounded jump intensity.

Application to birth-death processes shows exponential decay rate based on Wasserstein curvature.

Abstract

In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes. We apply the formula to estimate the exponential decrease rate of the Wasserstein distance between the marginals of two birth and death processes with the same generator in terms of the Wasserstein curvature.