The Recurrence Structure of General Markov Processes

We study the recurrence properties of a Markov process (Xt) with transition semigroup {Pt(x, A)} in terms of the properties of the associated Markov chains with one-step transition probabilities K F ( x , A ) = ∫ P t ( x , A ) F ( d t ) where F is a distribution on [0, ∞) with finite mean. We show that if (Xt) is a Hunt process, then (Xt) and KF have the same recurrence structures if F has some convolution power non-singular with respect to Lebesgue measure; and if Pt is continuous in t this extends to lattice F, and to arbitrary F if the continuity is uniform in the neighbourhoods of infinity. This extends and unifies known results for resolvent chains (F exponential) and skeleton chains (F concentrated at h > 0). Using these results, the second part of the paper finds topological conditions on the kernels KF which ensure that (Xt) admits a decomposition into a countable number of recurrent sets and a transient part of the space. These conditions are analogues of those previously shown to hold for discrete time Markov chains.