Distribution-Dependent Stochastic Functional Differential Equations

TL;DR

This paper proves existence and uniqueness for a class of distribution-dependent stochastic functional differential equations and establishes Harnack inequalities for their associated semigroups, advancing the theoretical understanding of DDSFDEs.

Contribution

It introduces an approximation method for DDSFDEs and derives Harnack inequalities for the related semigroup, extending classical results to distribution-dependent cases.

Findings

Existence and uniqueness of solutions for DDSFDEs established.

Harnack and shift-Harnack inequalities derived for the semigroup.

Extension of classical inequalities to distribution-dependent stochastic equations.

Abstract

By the approximation method introduced in \cite{FYW}, the existence and uniqueness are proved for a class of distribution-dependent stochastic functional differential equations (DDSFDEs). Moreover, combining the Harnack and shift-Harnack inequalities for classical stochastic functional differential equations with Girsanov's theorem, Harnack and shift-Harnack inequalities are obtained for the non-linear semigroup $P_t^*$ associated to the functional solution.