Large Deviation Principle for a Stochastic Heat Equation With Spatially Correlated Noise

Abstract

In this paper we prove a large deviation principle (ldp) for a perturbed stochastic heat equation defined on [0, T ] × [0, 1] d . This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the Hölder continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies a ldp and some requirements on the skeleton of the solution. Finally, we prove the called Freidlin-Wentzell inequality. In order to obtain all these results we need precise estimates of the fundamental solution of this equation.

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

“…the regularity of the probability measure induced by the solution [26,24,31], large deviation principles [25,19], Varadhan estimates [20,30], support theorems [21,16], path properties such as Hölder continuity [29,12] and much more. See also the references in these works for a more detailed account.…”

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.…”

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

“…For the case of wave equation in spatial dimension d = 3, the same method has been applied to derive an LDP result in [29], and also in several subsequent papers, for instance see [19,36]. The case of heat equation, governed by the same noise, has been considered in [24] taking the spatial coordinate x ∈ [0, 1] d , d ≥ 1 where the authors needs to establish precise estimates of the fundamental solution in order to obtain a Freidlin-Wentzell type inequality. In contrast, the approach we take in this paper is different from that of [24], and it is based on weak convergence arguments.…”

“…The case of heat equation, governed by the same noise, has been considered in [24] taking the spatial coordinate x ∈ [0, 1] d , d ≥ 1 where the authors needs to establish precise estimates of the fundamental solution in order to obtain a Freidlin-Wentzell type inequality. In contrast, the approach we take in this paper is different from that of [24], and it is based on weak convergence arguments. We can also refer to original LDP result for the case of a one-dimensional heat equation driven by a Brownian sheet in [10].…”