Large deviations for a class of stochastic partial differential equations

Kallianpur, G.; Xiong, Jie

doi:10.1214/aop/1042644719

Cited by 41 publications

(36 citation statements)

References 12 publications

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“…The ldp for spdes has been studied by many people (e.g. [2,[6][7][8]10,15,22,23,25]) but under a different framework to that used here. However, there are very few results on the ldp for nonlinear stochastic system with two widely separated timescales, as often appears in a complex system.…”

Section: Introductionmentioning

confidence: 99%

Large deviations and approximations for slow–fast stochastic reaction–diffusion equations

Wang

Roberts

Duan

2012

Journal of Differential Equations

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A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

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Section: Introductionmentioning

confidence: 99%

Large deviations and approximations for slow–fast stochastic reaction–diffusion equations

Wang

Roberts

Duan

2012

Journal of Differential Equations

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“…A second contribution is to show how the setup of [3], which considered SDEs driven by a Hilbert space valued Wiener processes, can be generalized to closely related settings, such as equations driven by a Brownian sheet. The chosen application is to a class of reaction-diffusion stochastic partial differential equations (SPDE) [see (5.1)], for which well-posedness has been studied in [20] and a small noise LDP established in [19]. The class includes, as a special case, the reaction-diffusion SPDEs considered in [25] (See Remark 3).…”

Section: (T) = B(x (T)) Dt + √ A(x (T)) Dw (T)mentioning

confidence: 99%

“…3. Lemma 4.1(ii) of [19] shows that, under Assumption 4, for any α <ᾱ there exists a constantK(α) such that, for all 0 ≤ t 1 ≤ t 2 ≤ T and all r 1 , r 2 ∈ O,…”

Section: Using (54) and (55) It Follows That For Any 0 ≤ S < T ≤ mentioning

confidence: 99%

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Large Deviations for Infinite Dimensional Stochastic Dynamical Systems

Budhiraja¹,

Dupuis²,

Maroulas³

2007

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The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.

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“…Most of the existing works [2,3,10,17], etc. essentially use the explicit formulas for the solution.…”

Section: Introductionmentioning

confidence: 99%

Small Perturbation of Stochastic Parabolic Equations: A Power Series Analysis

Lototsky

2002

Journal of Functional Analysis

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A semi-linear second-order stochastic parabolic equation is considered with coefficients, free terms, and initial condition depending on a parameter. It is shown that under some natural conditions the solution can be written as a power series in the parameter. The equations for the coefficients in the power series expansion are derived and the convergence of the power series is studied. An example from nonlinear filtering of diffusion process is discussed. # 2002 Elsevier Science (USA)

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Large deviations for a class of stochastic partial differential equations

Cited by 41 publications

References 12 publications

Large deviations and approximations for slow–fast stochastic reaction–diffusion equations

Large deviations and approximations for slow–fast stochastic reaction–diffusion equations

Large Deviations for Infinite Dimensional Stochastic Dynamical Systems

Small Perturbation of Stochastic Parabolic Equations: A Power Series Analysis

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