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

Abstract

Consider a Markov process $\{\Phi (t):t\ge 0\}$ evolving on a Polish space $\mathsf{X}$. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\psi$-irreducible and aperiodic. For a given function $f:\mathsf{X}:\to [1,\infty )$, under suitable conditions on the process the following are equivalent: \begin{enumerate} \item[(i)] There is a unique invariant probability measure $\pi$ satisfying $\int fd\pi <\infty$. \item[(ii)] There is a closed set $C$ satisfying $\psi (C)>0$ that is ``self $f$-regular.'' \item There is a function $V:\mathsf{X}\to (0,\infty ]$ that is finite on at least one point in $\mathsf{X}$, 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 $\mathcal{D}$ 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 $x\in \mathsf{X}$ satisfying $V(x)<\infty$, and any function $g:\mathsf{X}\to \Re$ for which $|g|$ is bounded by $f$, \[ \lim_{t\to\infty} {\sf E}_x[g(Φ(t))] = \int g\,dπ. \] 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 $g$.