THE QUASI-SURE RATIO ERGODIC THEOREM

Let (P-t)(t greater than or equal to 0) be the transition semigroup of a right Markov process, and let m be a conservative (P-t)-invariant m easure. Let f and g be elements of L-1(m) with g > 0. We show that, wi th the exception of an m-polar set of starting points x, the ratio int egral(0)(t) P(s)f(x) ds/ integral(0)(t) P(s)g(x) ils converges as t -- > +infinity, and we identify the limit as a ratio of conditional expec tations with respect to the appropriate invariant sigma-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 Ne veu's presentation of the Chacon-Ornstein filling scheme, adapted to c ontinuous time. The method yields, as a by-product, a local limit theo rem for the ratio of the ''characteristics'' of two continuous additiv e functionals, extending a result of G. Mokobodzki. (C) Elsevier, Pari s.