  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.
