A Bismut-Elworthy inequality for a Wasserstein diffusion on the circle

TL;DR

This paper develops a method to derive gradient estimates for infinite-dimensional Wasserstein diffusions on the circle, providing a Bismut-Elworthy-Li formula and analyzing the rate of blow-up of the gradient.

Contribution

It introduces a novel strategy to establish gradient estimates for Wasserstein space diffusions, including a Bismut-Elworthy-Li formula with explicit rate analysis.

Findings

Derived a Bismut-Elworthy-Li formula for Wasserstein diffusions on the circle

Established a gradient estimate with a blow-up rate of order O(t^{-(2+ε)})

Provided a new approach for infinite-dimensional diffusion analysis in Wasserstein spaces

Abstract

We introduce in this paper a strategy to prove gradient estimates for some infinite-dimensional diffusions on $L_2$-Wasserstein spaces. For a specific example of a diffusion on the $L_2$-Wasserstein space of the torus, we get a Bismut-Elworthy-Li formula up to a remainder term and deduce a gradient estimate with a rate of blow-up of order $\mathcal{O}(t^{-(2+\epsilon)})$.