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

2015/12/01 by Ioannis Kontoyiannis, Sean Meyn, Kontoyiannis, I. +1
Mathematics · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1512.00523

Abstract

Consider a Markov process \Φ(t) : t≥ 0\ evolving on a Polish space \sf 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\colon\sf X:→[1,∞), under suitable conditions on the process the following are equivalent: \beginenumerate \item[(i)] There is a unique invariant probability measure π satisfying ∫ f dπ0 that is ``self f-regular.'' \item There is a function V\colon\sf X → (0,∞] that is finite on at least one point in \sf X, for which the following Lyapunov drift condition is satisfied, \cal D V≤ - f+b\fieldI_C , \eqno\hbox(V3) where C is a closed small set and \cal D is the extended generator of the process. \endenumerate 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∈\sf X satisfying V(x)

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