On the $f$-Norm Ergodicity of Markov Processes in Continuous Time

TL;DR

This paper establishes a version of the $f$-Norm Ergodic Theorem for continuous-time Markov processes on Polish spaces, linking invariant measures, Lyapunov conditions, and ergodic convergence under certain regularity assumptions.

Contribution

It extends the well-known discrete-time ergodic results to continuous-time processes, providing conditions for uniqueness of invariant measures and ergodic convergence using Lyapunov functions.

Findings

Equivalent conditions for invariant measure existence and uniqueness.

Lyapunov drift conditions imply ergodic convergence.

Extension of discrete-time ergodic theorems to continuous-time processes.

Abstract

Consider a Markov process $\{\Phi(t) : t\geq 0\}$ evolving on a Polish space ${\sf X}$. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\psi$-irreducible and aperiodic. For a given function $f\colon{\sf X}:\to[1,\infty)$, under suitable conditions on the process the following are equivalent: \begin{enumerate} \item[(i)] There is a unique invariant probability measure $\pi$ satisfying $\int f\,d\pi<\infty$. \item[(ii)] There is a closed set $C$ satisfying $\psi(C)>0$ that is ``self $f$-regular.'' \item There is a function $V\colon{\sf X} \to (0,\infty]$ that is finite on at least one point in ${\sf X}$, for which the following Lyapunov drift condition is satisfied, \[ {\cal D} V\leq - f+b\field{I}_C\, , \eqno{\hbox{(V3)}} \] where $C$ is a closed small set and ${\cal D}$ is the extended generator of the process. \end{enumerate} For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of $\bfPhi$ under a suitable norm is also obtained: For each initial condition $x\in{\sf X}$ satisfying $V(x)<\infty$, and any function $g\colon{\sf X}\to\Re$ for which $|g|$ is bounded by $f$, \[ \lim_{t\to\infty} {\sf E}_x[g(\Phi(t))] = \int g\,d\pi. \] Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process $\{\Phi(t)\}$ or on the function $g$.