A Ruelle Operator for continuous time Markov Chains

Resumo

We consider a generalization of the Ruelle theorem for thecase of continuous time problems. We present a result which we believeis important for future use in problems in Mathematical Physics relatedto C-Algebras.We consider a nite state set S and a stationary continuous timeMarkov Chain Xt, t 0, taking values on S. We denote by  the setof paths w taking values on S (the elements w are locally constant withleft and right limits and are also right continuous on t). We consideran innitesimal generator L and a stationary vector p0. We denoteby P the associated probability on (; B). All functions f we considerbellow are in the set L1(P).From the probability P we dene a Ruelle operator Lt; t 0, actingon functions f :  ! R of L1(P). Given V :  ! R, such that isconstant in sets of the form fX0 = cg, we dene a modied Ruelleoperator ~ LtV ; t 0, in the following way: there exist a certain fV suchthat for each t we consider the operator acting on g given by~ LtV (g)(w) = [1fVLt(eR t0 (V s)(:)ds g fV ) ] (w)We are able to show the existence of an eigenfunction u and aneigen-probability V on  associated to ~ LtV ; t 0.We also show the following property for the probability V : for anyintegrable g 2 L1(P) and any real and positive tZe