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

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 $g$.