Large deviations for a fractional stochastic heat equation in spatial dimension d, driven by a spatially correlated noise

Mohamed Mellouk (1) , Tarik El Mellali

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

In this paper we study the Large Deviations Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space Rd, with arbitrary dimension d ≥ 1, under random influence which is a Gaussian noise, white in time and correlated in space. The differential operator is a fractional derivative operator. We prove a Large deviations principle for our equation, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion. This approach reduces the proof of LDP to establishing basic qualitative properties for controlled analogues of the original stochastic system.

Mots clés: : Fractional derivative operator; stochastic partial differential equation; correlated Gaussian noise; Fourier transform; large deviations principle; weak convergence method.

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Mohamed Mellouk, Tarik El Mellali. Large deviations for a fractional stochastic heat equation in spatial dimension d, driven by a spatially correlated noise : . Stochastics and Dynamics, 2016, Vol. 16, No. 1 (2016), ⟨10.1142/S0219493716500015⟩. ⟨hal-01437848⟩