1 Harris recurrence This introductory section states some main facts about Harris recurrence. These facts will be used througout this text. An essential reference is Azema- Duflo-Revuz [A-D-R 69]. Throughout this note, we consider a stochastic basis (Q,A,1F), IF right- continuous, and on (ft, A, IF, (Px)x) a process X = (Xt)to which is strongly Markov, taking values in a Polish space (E,£), with cadlag paths, and with A^o = x P^-a.s., x E E. We have shift operators (#t)to on (fi,„4,iF), and write (Pt)to for the semigroup of X. 1.1 Definition: ([A-D-R 69]) X is called Harris recurrent if there exists some cr-finite measure m on (E,£) such that (*) m(A)0 = VxeE: Px ( f lA(Xs)ds = oo) =1. Sometimes also the terminology m-irreducible is used for (*). 1.2 Theorem: ([A-D-R 69]) If the process X is Harris recurrent, then a uni- que (up to constant multiples) invariant measure /i for X exists (i.e. a cr-finite measure such that \iPt \x for all t 0), and property (*) in 1.1 holds with ji in place of m. Definition: A Harris recurrent process X with invariant measure /i is called positive recurrent (or ergodic) ii fi(E) oo, null recurrent if ji(E) = oo. We give the major ideas of the proof of 1.2 the notions developped here will reappear as main tools in section 7. Sketch of the proof of 1.2: For a 0, the a-potential kernel is poo roo Ua(x,A) = Ex{ / e-atl A (Xt)dt) = / e-atP t (x,A)dt Jo Jo i) A first step is to prove that \xPt = \i for all t if and only if \i\Jx = \x. The nontrivial direction is =. The proof starts from the resolvent equation (see [Chu 82, p.83]) Ua = Up + ((3 - a)UaU0 = Uf} + (P- ajUPU" for all a 0, j3 0. In particular for a 1 U1 = Ua + U1Ia^Ua 7
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