$L^p$ properties for Gaussian random series
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- by Antoine Ayache and Nikolay Tzvetkov PDF
- Trans. Amer. Math. Soc. 360 (2008), 4425-4439 Request permission
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 $\mathbb R^2$.References
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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
- Received by editor(s): October 3, 2006
- Published electronically: March 12, 2008
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 360 (2008), 4425-4439
- MSC (2000): Primary 35Q55, 37K05, 37L50, 60G15, 60G50
- DOI: https://doi.org/10.1090/S0002-9947-08-04456-5
- MathSciNet review: 2395179