$L^p$ properties for Gaussian random series

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 $\sup_{n\in\mathbb{N}^\star}\Vert e_n\Vert _{L^p(Y,{\mathcal M},\mu)}<\infty$. 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 $\mathbb{R}^2$.