Poisson-Orlicz norm and infinite Ergodic Theory

Abstract

Urbanik's theorem for a Poisson process on an infinite measure space (X, A, $\mu$) relates integrability of stochastic integrals to a particular Orlicz function space L$\Phi$ ($\mu$) on which the L1-norm of the Poisson process induces a norm (called Poisson-Orlicz in the sequel) that is shown to be equivalent to the classical gauge and Orlicz norms.We obtain a full characterization of stochastic integrals using difference operators that, together with a simple duality argument, allows to derive Urbanik's theorem as well as an optimal inequality between the Orlicz and the Poisson-Orlicz norm.In a second part, we show that the Poisson-Orlicz norm plays a role in infinite Ergodic Theory where it is seen as an alternative to the L1-norm to identify several dynamical invariants that the latter fails to identify. We also show that, whereas the L1-norm fully characterizes exact endomorphisms (Lin's theorem), Poisson-Orlicz norm fully characterizes remotely infinite endomorphisms.