Quantitative sensitivity analysis for Fokker-Planck equation with respect to the Wasserstein distance

TL;DR

This paper provides quantitative bounds on how solutions to the Fokker-Planck equation change with parameters, using Wasserstein distances, and applies these results to Langevin processes.

Contribution

It introduces two methods for bounding the Wasserstein distance between solutions with different parameters, enhancing understanding of solution sensitivity.

Findings

Established bounds for Wasserstein distances between solutions

Provided two proof techniques: coupling and duality differentiation

Applied results to Langevin processes for convergence and sensitivity analysis

Abstract

We analyze the sensitivity of solutions to the Fokker-Planck equation with respect to some unknown parameter. Our main result is to provide quantitative upper bounds for the $p$-Wasserstein distance $\mathcal{W}_p$ between two solutions with different parameters, for every $p \geq 2$. We are able to give two proofs of this result, the first relying on synchronous coupling between two solutions of an SDE, and another one that relies on the differentiation of Kantorovitch dual formulation of optimal transport. We also provide more specific bounds in the case of the overdamped Langevin process, for which we are able to compare convergence to the invariant measure and sensitivity to the parameter.