Abstract
In this paper we prove a large deviation principle (LDP) for a perturbed stochastic heat equation defined on $[0,T]\times [0,1]^d$. This equation is driven by a Gaussian noise, white in time and correlated in space. Firstly, we show the Holder continuity for the solution of the stochastic heat equation. Secondly, we check that our Gaussian process satisfies an 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.
Highlights
Consider the following perturbed d-dimensional spatial stochastic heat equation on the compact set [0, 1]dLuε(t, x) = εα(uε(t, x))F (t, x) + β(uε(t, x)), t ≥ 0, x ∈ [0, 1]d, uε(t, x) = 0, x ∈ ∂([0, 1]d), uε(0, x)= 0, x ∈ [0, 1]d, (1.1) with ε > L = ∂ ∂t −∆ where
This means that we check the existence of a lower semi-continuous function I : Cγ,γ(DTd ) → [0, ∞], called rate function, such that {I ≤ a} is compact for any a ∈ [0, ∞), and ε2 log P {uε ∈ O} ≥ −Λ(O), for each open set O, ε2 log P {uε ∈ U } ≤ −Λ(U ), for each closed set U, where, for a given subset A ∈ Cγ,γ(DTd ), Λ(A) = inf I(l)
We will follow the approach of Freidlin and Wentzell [10] for diffusion process
Summary
Consider the following perturbed d-dimensional spatial stochastic heat equation on the compact set [0, 1]d. Under (C) and (Hη) for some η ∈ (0, 1), we will prove the most important result of this paper: the existence of a large deviation principle (ldp) for the law of the solution uε to on This means that we check the existence of a lower semi-continuous function I : Cγ,γ(DTd ) → [0, ∞], called rate function, such that {I ≤ a} is compact for any a ∈ [0, ∞), and ε2 log P {uε ∈ O} ≥ −Λ(O), for each open set O, ε2 log P {uε ∈ U } ≤ −Λ(U ), for each closed set U, where, for a given subset A ∈ Cγ,γ(DTd ), Λ(A) = inf I(l). We will follow the approach of Freidlin and Wentzell [10] for diffusion process (see Dembo and Zeitouni [6]) Another remarkable article is Chenal and Millet [3] where they prove the existence of a ldp for a one-dimensional stochastic heat equation.
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