Exponential Mixing for Retarded Stochastic Differential Equations

TL;DR

This paper establishes exponential mixing properties for various classes of retarded stochastic differential equations, including non-autonomous, neutral, and jump-diffusion types, using advanced probabilistic and topological methods.

Contribution

It provides new results on exponential mixing for retarded SDEs with different features, employing novel techniques like Arzelà–Ascoli, Razumikhin, and Kurtz criteria.

Findings

Exponential mixing proven for non-autonomous retarded SDEs

Exponential mixing established for neutral SDEs with continuous paths

Exponential mixing demonstrated for jump-diffusion retarded SDEs

Abstract

In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded SDEs by the Arzel\`{a}--Ascoli tightness characterization of the space $\C$ equipped with the uniform topology (b) neutral SDEs with continuous sample paths by a generalized Razumikhin-type argument and a stability-in-distribution approach and (c) jump-diffusion retarded SDEs by the Kurtz criterion of tightness for the space $\D$ endowed with the Skorohod topology.