Bismut Formula for Lions Derivative of Distribution Dependent SDEs and Applications

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.