POISSON-ORLICZ NORM IN INFINITE ERGODIC THEORY

Starting with an infinite measure Lebesgue space (X, A, µ), we consider two norms on L 1 (µ) that are both weaker than • 1. The first one is associated to a particular Orlicz space and allows to obtain general ergodic theorems for a measure preserving transformation, involving the convergence of its Birkhoff sums, that are true for L p (µ) , • p , 1 < p < ∞, but fail for p = 1 as the measure is infinite. With the second, we obtain non-trivial T-invariant vectors in the completion of L 1 (µ) with respect to this new norm and this leads to a L 1 (µ)-characterization of ergodicity. Both norms are in fact related to the Poisson process over (X, A, µ) and their family of stochastic integrals whose inherent integrability constraints are the main ingredient in the proofs. A new characterization of stochastic integrals is obtained and additional results on coboundaries come out as a natural byproduct