Existence of Invariant Probability Measures for Stochastic Differential Equations with Finite Time Delay

TL;DR

This paper establishes conditions for the existence of invariant probability measures in stochastic delay differential equations, using the Krylov-Bogoliubov method, and applies these results to systems driven by Lévý noise.

Contribution

It provides new sufficient conditions for invariant measure existence in stochastic delay equations, including cases with Lévý noise and boundedness in probability.

Findings

Invariant measures exist under certain boundedness conditions.

Boundedness in probability of solutions implies boundedness of solution segments.

Applications to stochastic Mackey-Glass and Wright's equations.

Abstract

We provide sufficient conditions for the existence of invariant probability measures for generic stochastic differential equations with finite time delay. This is achieved by means of the Krylov-Bogoliubov method. Furthermore, we focus on stochastic delay equations whose deterministic coefficient satisfies a one-sided bound, which enables us to show that boundedness in probability of a solution $X(t)$ entails boundedness in probability of its solution segment $X_t$. This implies that for a large set of systems, we can infer that an invariant measure exists if only there is at least one solution that is bounded in probability. Applications include, but are not limited to, the stochastic Mackey-Glass equations and the stochastic Wright's equation. The noise driving the dynamical system is allowed to be an integrable L\'evy process.