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

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 Xt. 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 [M. van den Bosch, O. W. van Gaans, and S. M. Verduyn Lunel, SIAM J. Appl. Dyn. Syst., 25 (2026)] and the stochastic Wright's equation [M. van den Bosch, O. W. van Gaans, and S. M. Verduyn Lunel, Stochastic Wright's Equation: Existence of Invariant Measures, preprint, arXiv:2605.09805, 2026]. The noise driving the dynamical system is allowed to be an integrable Lévy process.