Bismut Formula and Gradient Estimates for Dirichlet Semigroups with Application to Singular Killed DDSDEs

TL;DR

This paper develops a local Bismut formula for Dirichlet semigroups to obtain gradient estimates for killed SDEs with singular drifts and applies these results to bound the total variation distance between solutions of killed DDSDEs.

Contribution

It introduces a local Bismut formula for Dirichlet semigroups and applies it to derive gradient estimates and bounds for solutions of killed DDSDEs with singular drifts.

Findings

Gradient estimates for killed SDEs with singular drifts

Bound on total variation distance via truncated Wasserstein distance

Extension of Bismut formula to local Dirichlet semigroups

Abstract

By establishing a local version of Bismut formula for Dirichlet semigroups on a regular domain, gradient estimates are derived for killed SDEs with singular drifts. As an application, the total variation distance between two solutions of killed DDSDEs is bounded above by the truncated $1$-Wasserstein distance of initial distributions, in the regular and singular cases respectively.