The recurrence structure of general Markov processes

We study properties of a Markov process (X) with transition semigroup {P'(x, A)} in terms of the properties of the associated Markov chains with one-step transition probabilities KF(x, A) =JP'(x, A)F(dt) where F is a distribution on [0,00) with finite mean. We show that if (X) is Hunt, then (X,) and KF have the same recurrence structures if F has some convolution power non-singular with respect to Lebesgue measure; and if P' is continuous in t this extends to lattice F, and to arbitrary F if the continuity is uniform in the neighbourhoods of infinity. Here, by recurrence structures we mean such properties as a given subset being a maximal closed set or a maxima\ Harris set, or a grven maximal c,\()~ed set being indecomposable and properly essential or being not properly essential. This extends and unifies known results for resolvent chains (F exponential) and skeleton chains (F concentrated at h > 0).