On the f-norm ergodicity of markov processes in continuous time

Consider a Markov process Φ = { Φ (t): t ≥ 0} evolving on a Polish space X. A version of the f -Norm Ergodic Theorem is obtained: Suppose that the process is ψ-irreducible and aperiodic. For a given function f: X → [1, ∞), under suitable conditions on the process the following are equivalent: ∫(i) There is a unique invariant probability measure π satisfying f dπ 0 that is “self f -regular.” (iii) There is a function V: X → (0, ∞] that is finite on at least one point in X, for which the following Lyapunov drift condition is satisfied, (Formula Presented), (V3) where C is a closed small set and D is the extended generator of the process. For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of Φ under a suitable norm is also obtained: For each initial condition x ∈ X satisfying V (x) < ∞, and any function g: X → R for which |g| is bounded by f, (Formula Presented) Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process Φ or on the function g