Strong Ratio Limit Theorems for Markov Processes

Let Pbe a conservative and ergodic Markov operator on L_\infty(X, \Sigma, m)(where m(X) = 1. It is proved that if for A \in \Sigmawith m(A) > 0and \mu \ll ma finite measure with \mu(A) > 0 \lim_{n\rightarrow\infty} < \mu, P^{n+1}1_B >/<\mu, P^n 1_A>exists for every B \subset A then Phas a \sigmafinite invariant measure \lambdaand there is a sequence A_k \uparrow Xwith A_0 = Asuch that for $0 \leqq f, g \in...