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

Abstract

In this paper we study the Large Deviation Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space R d , 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.

Moderate deviations for a fractional stochastic heat equation with spatially correlated noise

“…Using the strategy in the proof of Proposition 2.7 in [7], one can obtain the following result. Here we omit its proof.…”

“…Also refer to the weak convergence approach used in [7]. By the Skorokhod representation theorem, there exist a probability pΩ,F , pF t q,Pq, and, on this basis, a sequence of independent Brownian motionsB " pB k q kě1 and also a family ofF t -predictable processes tv ε ; ε ą 0u,v belonging to L 2 pΩˆr0, T s; Hq taking values on H N T ,P-a.s., such that the joint law of pv ε , v, Bq under P coincides with that of pv ε ,v,Bq underP and lim εÑ0 xv ε´v , gy H T " 0, @g P H T ,P´a.s.. …”

Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise

“…Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise. Marquez-Carreras and Sarra [3] proved a LDP for a stochastic heat equation with spatially correlated noise, and Mellali and Mellouk [4] extended Marquez-Carreras and Sarra's [3] to a fractional operator. Jiang et al [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise.…”

“…Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models (such as interacting particle systems) (see [2][3][4][5][6][7]). Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise.…”

Central limit theorem for a fractional stochastic heat equation with spatially correlated noise

“…• The case λ(ε) = 1/ √ ε provides some large deviation estimates. El Mellali and Mellouk [7] proved that the law of the solution u ε satisfies a large deviation principle.…”