Exponential ergodicity for retarded stochastic differential equations

Abstract

This paper establishes ergodic properties for Markovian semigroups generated by segment processes associated with several classes of retarded stochastic differential equations (SDEs) with constant/variable/distributed time lags. It derives exponential ergodicity for (a) retarded SDEs by the Arzelà–Ascoli tightness characterization of the space 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) retarded SDEs driven by jump processes using the Kurtz criterion of tightness for the space endowed with the Skorohod topology.

Keywords:

• retarded stochastic differential equation
• invariant measure
• exponential ergodicity
• unform metric
• Skorohod metric

AMS Subject Classifications:

• 60H15
• 60J25
• 60H30
• 39B82

Notes

This research was supported in part by the Army Research Office under [grant number W911NF-12-1-0223].