Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity

TL;DR

This paper establishes the existence and uniqueness of stationary distributions for retarded stochastic differential equations without requiring dissipative conditions, using a variation-of-constants approach to handle pure delay systems.

Contribution

It introduces a novel method to prove stationary distributions for delay SDEs lacking dissipativity, including neutral type and Lévy-driven equations.

Findings

Proves existence and uniqueness of stationary distributions for a broad class of delay SDEs.

Handles systems driven by Lévy processes without finite second moments.

Extends results to neutral type and pure delay systems without dissipative assumptions.

Abstract

Retarded stochastic differential equations (SDEs) constitute a large collection of systems arising in various real-life applications. Most of the existing results make crucial use of dissipative conditions. Dealing with "pure delay" systems in which both the drift and the diffusion coefficients depend only on the arguments with delays, the existing results become not applicable. This work uses a variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. This paper establishes existence and uniqueness of stationary distributions for retarded SDEs that need not satisfy dissipative conditions. The retarded SDEs considered in this paper also cover SDEs of neutral type and SDEs driven by L\'{e}vy processes that might not admit finite second moments.