Bismut Formula for Lions Derivative of Distribution Dependent SDEs and Applications

TL;DR

This paper develops Bismut formulas for the Lions derivative of distribution-dependent SDEs using Malliavin calculus, providing explicit estimates for derivatives and distribution distances in both degenerate and non-degenerate cases.

Contribution

It introduces a novel Bismut formula for Lions derivatives of distribution-dependent SDEs, overcoming key difficulties due to the absence of semigroup properties.

Findings

Explicit estimates for Lions derivatives

Bounds on total variational distance between distributions

Applicability to both degenerate and non-degenerate cases

Abstract

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of $P_tf(\mu):=\mathbb E f(X_t^\mu)$, where $t>0,$ $ f $ is a bounded measurable function, and $X_t^\mu$ solves a distribution dependent SDE with initial distribution $\mu$. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula $P_tf (\mu)= \int P_tf(x)\mu(d x)$, essential difficulties are overcome in the study.