Bismut Formula for Intrinsic Derivative of DDSDEs with Singular Interactions

TL;DR

This paper extends Bismut formulas to the intrinsic derivatives of distribution-dependent stochastic equations with singular interactions, advancing the mathematical understanding of such complex systems.

Contribution

It introduces Bismut type formulas for DDSDEs with singular interactions, generalizing previous results that required differentiable drifts.

Findings

Established Bismut formulas for intrinsic derivatives of DDSDEs with singular interactions.

Extended existing formulas beyond the case with Lion's differentiable drifts.

Contributed to the theoretical framework for analyzing DDSDEs with singularities.

Abstract

In recent years, remarkable progress has been made for Distribution dependent stochastic equations (DDSDEs) with singular interactions, existing results include wellposedness, propagation of chaos, entropy cost inequality and ergodicity. As a continuation to the existing study, in this paper we establish Bismut type formulas for the intrinsic derivative of DDSDEs with singular interactions, which extends the existing formula established for the case with Lion's differentiable drifts.