Invariant Measure for Neutral Stochastic Functional Differential Equations with Non-Lipschitz Coefficients

TL;DR

This paper investigates the long-term behavior of neutral stochastic functional differential equations with non-Lipschitz nonlinearities in Hilbert spaces, establishing the existence of invariant measures using the Krylov-Bogoliubov theorem.

Contribution

It introduces a method to prove the existence of invariant measures for a class of complex stochastic differential equations with non-Lipschitz nonlinearities.

Findings

Existence of invariant measures in shift spaces for the equations.

Application of Krylov-Bogoliubov theorem to neutral stochastic equations.

Extension to equations with non-Lipschitz nonlinearities.

Abstract

In this work we study the long time behavior of nonlinear stochastic functional-differential equations of neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish the existence of invariant measures in the shift spaces for such equations. Our approach is based on Krylov-Bogoliubov theorem on the tightness of the family of measures.