Lp properties for Gaussian random series

Let c = (c(n))(n epsilon N*) be an arbitrary sequence of l(2)(N*) and let F-c(omega) be a random series of the type F-c(omega) = Sigma(gn)(n epsilon N*)(omega)c(n)e(n), where (g(n))(n epsilon N*) is a sequence of independent N-C(0, 1) Gaussian random variables and (e(n))(n epsilon N*) an orthonormal basis of L-2(Y, M, mu) (the finite measure space (Y, M, mu) being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and su. cient condition for F-c(omega) to belong to L-p(Y, M, mu), p epsilon [2, infinity) for any c epsilon l(2)(N*) almost surely is that sup(n epsilon N*) parallel to e(n)parallel to(Lp(Y, M, mu)) < infinity. 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 Schrodinger equation posed on the open unit disc of R-2.