Existence of Periodic and Stationary Solutions to Distribution-Dependent SDEs

TL;DR

This paper develops new criteria using weak convergence and Lyapunov functions to establish the existence of periodic and stationary solutions in distribution-dependent stochastic differential equations, supported by concrete examples.

Contribution

It introduces novel criteria for solutions existence in distribution-dependent SDEs by combining weak convergence and Lyapunov methods.

Findings

Criteria successfully identify periodic solutions

Criteria successfully identify stationary solutions

Concrete examples validate the criteria

Abstract

We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and Lyapunov functions can be combined to give efficient criteria for the existence of periodic and stationary solutions. Concrete examples are presented to illustrate the novel criteria.