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 $g$.
Categories: math.PR