L Properties for Gaussian Random Series

Let c = (cn)n∈N be an arbitrary sequence of l2(N ) and let Fc(ω) be a random series of the type Fc(ω) = ∑ n∈N gn(ω)cnen, where (gn)n∈N∗ is a sequence of independent NC(0, 1) Gaussian random variables and (en)n∈N an orthonormal basis of L2(Y,M, μ) (the finite measure space (Y,M, μ) being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for Fc(ω) to belong to Lp(Y,M, μ), p ∈ [2,∞) for any c ∈ l2(N ) almost surely is that supn∈N ‖en‖Lp(Y,M,μ) < ∞. One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrödinger equation posed on the open unit disc of R2.