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

Feng-Yu Wang, Xiao-Yu Zhao

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.

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

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 -Wasserstein distance of initial distributions, in the regular and singular cases respectively.

Paper Structure

This paper contains 9 sections, 6 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Bismut formula and gradient estimate
  3. Bismut formula in the regular case
  4. Gradient estimate in the singular case
  5. Gradient estimate for killed DDSDEs
  6. Well-posedness of \ref{['E1']}
  7. Gradient estimate in the regular case
  8. Gradient estimate in the singular case
  9. Acknowledgement.

Key Result

Theorem 2.1

Assume (B). For $(t,x)\in (0,T]\times D$, let $(\beta_s)_{s\in [0,t]}$ be an adapted absolutely continuous real process such that and for some $\varepsilon>0$ then for any $v\in \mathbb R^d$ and $f\in \mathscr B_b(\bar{D})$, Consequently, for any $p\in (1+\varepsilon^{-1},\infty]$, there exists a constant $c(p,\varepsilon)>0$ such that for any $t\in(0,T],\,x\in D$ and $f\in \mathscr B_b(\bar{D}

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['T1']}
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 4 more