On the $f$-norm ergodicity of Markov processes in continuous time

Abstract

Consider a Markov process $\boldsymbol{\Phi } =\{ \Phi (t) : t\geq 0\}$ 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 $f\colon \mathsf{X} \to [1,\infty )$, under suitable conditions on the process the following are equivalent:

(i) There is a unique invariant probability measure $\pi $ satisfying $\int f\,d\pi <\infty $.

(ii) There is a closed set $C$ satisfying $\psi (C)>0$ that is “self $f$-regular.”

(iii) There is a function $V\colon \mathsf{X} \to (0,\infty ]$ that is finite on at least one point in $\mathsf{X} $, for which the following Lyapunov drift condition is satisfied, \[ \mathcal{D} V\leq - f+b\mathbb{I}_C\, , \tag{V3} \] where $C$ is a closed small set and $\mathcal{D}$ is the extended generator of the process.

For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of $\boldsymbol{\Phi }$ under a suitable norm is also obtained: For each initial condition $x\in \mathsf{X} $ satisfying $V(x)<\infty $, and any function $g\colon \mathsf{X} \to \mathbb{R} $ for which $|g|$ is bounded by $f$, \[ \lim _{t\to \infty } \mathsf{E}_x[g(\Phi (t))] = \int g\,d\pi . \] Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process $\boldsymbol{\Phi } $ or on the function $g$.