Ergodic properties of generalized Ornstein--Uhlenbeck processes

TL;DR

This paper studies the ergodic behavior of generalized Ornstein--Uhlenbeck processes, establishing conditions for ergodicity and convergence rates using Foster--Lyapunov methods and generator analysis.

Contribution

It provides new sufficient conditions for ergodicity and convergence rates of generalized Ornstein--Uhlenbeck processes, including cases of optimality.

Findings

Conditions for ergodicity and convergence rates are established.

Explicit drift conditions are derived from the generator.

Optimality of results is demonstrated in special cases.

Abstract

We investigate ergodic properties of generalized Ornstein--Uhlenbeck processes. In particular, we provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster--Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.