Bismut Formula for Lions Derivative of Distribution Dependent SDEs and Applications

Abstract

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of $P_{t}f(\mu ):=\mathbb{E}f(X_t^{\mu })$, 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 $P_{t}f(\mu )=\int P_{t}f(x)\mu (dx)$, essential difficulties are overcome in the study.

31 pages

References in corpus (1)

• Formulae for the derivatives of heat semigroups

Cited by in corpus (2)

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