The quasi-sure ratio ergodic theorem

Abstract

Let ( P t ) t≥o be the transition semigroup of a right Markov process, and let m be a conservative ( P t )-invariant measure. Let ƒ and g be elements of L 1( m) with g > 0. We show that, with the exception of an m-polar set of starting points x, the ratio ∫ 0 t P sƒ ( x) ds/ ∫ 0 t P sg ( x) ds converges as t → +∞, and we identify the limit as a ratio of conditional expectations with respect to the appropriate invariant σ-algebra. This improves upon earlier work of M. Fukushima and M.G. Shur, in which the exceptional set was shown to be m-semipolar. The proof is based on Neveu's presentation of the Chacon-Ornstein filling scheme, adapted to continuous time. The method yields, as a by-product, a local limit theorem for the ratio of the “characteristics” of two continuous additive functionals, extending a result of G. Mokobodzki.