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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ L^p$ properties for Gaussian random series


Authors: Antoine Ayache and Nikolay Tzvetkov
Journal: Trans. Amer. Math. Soc. 360 (2008), 4425-4439
MSC (2000): Primary 35Q55, 37K05, 37L50, 60G15, 60G50
Published electronically: March 12, 2008
MathSciNet review: 2395179
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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

$\displaystyle 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$.


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Additional Information

Antoine Ayache
Affiliation: Laboratoire Paul Painlevé, Bât. M2, Université Lille 1, 59 655 Villeneuve d’Ascq Cedex, France
Email: antoine.ayache@math.univ-lille1.fr

Nikolay Tzvetkov
Affiliation: Laboratoire Paul Painlevé, Bât. M2, Université Lille 1, 59 655 Villeneuve d’Ascq Cedex, France
Email: nikolay.tzvetkov@math.univ-lille1.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04456-5
PII: S 0002-9947(08)04456-5
Keywords: Eigenfunctions, Gaussian random series
Received by editor(s): October 3, 2006
Published electronically: March 12, 2008
Article copyright: © Copyright 2008 American Mathematical Society