POISSON-ORLICZ NORM AND INFINITE ERGODIC THEORY

Emmanuel Roy (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

Urbanik’s theorem for a Poisson process on an infinite measure space (X, A, μ) relates integrability of stochastic integrals to a particular Orlicz function space LΦ (μ) 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.

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Emmanuel Roy. POISSON-ORLICZ NORM AND INFINITE ERGODIC THEORY. 2023. ⟨hal-03362759v2⟩