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

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.