On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact riemannian manifold

Zdzislaw Brzezniak (1) , Annie Millet (2, 3)

1 University of York [York, UK]
2 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We prove the existence and the uniqueness of a solution to the stochastic NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a recent work by Burq, Gérard and Tzvetkov in the deterministic setting, and a series of papers by de Bouard and Debussche, who have examined similar questions in the case of the flat euclidean space with random perturbation. We prove the existence and the uniqueness of a local maximal solution to stochastic nonlinear Schrödinger equations with multiplicative noise on a compact d-dimensional riemannian manifold. Under more regularity on the noise, we prove that the solution is global when the nonlinearity is of defocusing or of focusing type, d=2 and the initial data belongs to the finite energy space. Our proof is based on improved stochastic Strichartz inequalities.

Mots clés: Non linear Schrödinger equation; stochastic Strichartz inequality; multiplicative noise; Radonifying Banach spaces; local solution; global solution; focusing; defocusing

Domaines: Probabilités [math.PR]

Zdzislaw Brzezniak, Annie Millet. On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact riemannian manifold. Potential Analysis, 2014, 41 (2), p. 269-315. ⟨hal-00732683⟩