$L^p$ properties for Gaussian random series

Abstract:

Let $c=(c_n)_{n\in \mathbb N^\star }$ be an arbitrary sequence of $l^2(\mathbb {N}^{\star })$ and let $F_c (\omega )$ be a random series of the type \[ F_c (\omega )=\sum _{n\in \mathbb N^\star }g_n (\omega ) c_n e_n , \] where $(g_n)_{n\in \mathbb N^*}$ is a sequence of independent ${\mathcal N}_{\mathbb C}(0,1)$ Gaussian random variables and $(e_n)_{n\in \mathbb N^\star }$ an orthonormal basis of $L^2(Y,{\mathcal M},\mu )$ (the finite measure space $(Y,{\mathcal M},\mu )$ 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 $F_c (\omega )$ to belong to $L^p(Y,{\mathcal M},\mu )$, $p\in [2,\infty )$ for any $c\in l^2 (\mathbb N^\star )$ almost surely is that $\sup _{n\in \mathbb N^\star }\|e_n\|_{L^p(Y,{\mathcal M},\mu )}<\infty$. 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 $\mathbb R^2$.