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Journal ArticleDOI

Moderate deviation principle for a class of stochastic partial differential equations

01 Mar 2016-Journal of Applied Probability (Applied Probability Trust)-Vol. 53, Iss: 1, pp 279-292
TL;DR: The moderate deviation principle is established for a class of stochastic partial differential equations with non-Lipschitz continuous coefficients and derived for two important population models: super-Brownian motion and the Fleming–Viot process.
Abstract: We establish the moderate deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, we derive the moderate deviation principle for two important population models: super-Brownian motion and the Fleming-Viot process.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors established a moderate deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative Levy noises and showed that the weak convergence method introduced by Budhiraja, Dupuis and Ganguly in [3] plays a key role.

37 citations

Posted Content
TL;DR: In this article, a moderate deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative $L\acute{e}vy$ noises is established.
Abstract: In this paper, we establish a moderate deviation principle for two-dimensional stochastic Navier-Stokes equations driven by multiplicative $L\acute{e}vy$ noises. The weak convergence method introduced by Budhiraja, Dupuis and Ganguly in arXiv:1401.73v1 plays a key role.

37 citations

Journal ArticleDOI
TL;DR: This is the first result about the limit theorem and the moderate deviations for stochastic primitive equations driven by multiplicative noise and the weak convergence method is established.
Abstract: In this paper, we establish a central limit theorem and a moderate deviation for two-dimensional stochastic primitive equations driven by multiplicative noise. This is the first result about the limit theorem and the moderate deviations for stochastic primitive equations. The proof of the results relies on the weak convergence method and some delicate and careful a p r i o r i estimates.

14 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the stochastic tidal dynamics equations perturbed by multiplicative Gaussian noise and discuss some asymptotic behaviors, including a central limit theorem and a moderate deviation.
Abstract: In this article, we consider the stochastic tidal dynamics equations perturbed by multiplicative Gaussian noise and discuss some asymptotic behaviors. A central limit theorem and a moderate deviati...

11 citations

Posted Content
TL;DR: In this paper, a large deviation principle for non-linear monotone stochastic partial differential equations is derived using the hyperexponential recurrence criterion for the occupation measure.
Abstract: Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises.

8 citations

References
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Book
27 Mar 1998
TL;DR: The LDP for Abstract Empirical Measures and applications-The Finite Dimensional Case and Applications of Empirically Measures LDP are presented.
Abstract: LDP for Finite Dimensional Spaces.- Applications-The Finite Dimensional Case.- General Principles.- Sample Path Large Deviations.- The LDP for Abstract Empirical Measures.- Applications of Empirical Measures LDP.

5,578 citations

Book
01 Jan 1997
TL;DR: The Laplace Principle for the Random Walk Model with Discontinuous Statistics as mentioned in this paper has been extended to the continuous-time Markov Processes with continuous statistics, and the Laplace principle has been used for the continuous time Markov Chain model as well.
Abstract: Formulation of Large Deviation Theory in Terms of the Laplace Principle. First Example: Sanov's Theorem. Second Example: Mogulskii's Theorem. Representation Formulas for Other Stochastic Processes. Compactness and Limit Properties for the Random Walk Model. Laplace Principle for the Random Walk Model with Continuous Statistics. Laplace Principle for the Random Walk Model with Discontinuous Statistics. Laplace Principle for the Empirical Measures of a Markov Chain. Extensions of the Laplace Principle for the Empirical Measures of a Markov Chain. Laplace Principle for Continuous-Time Markov Processes with Continuous Statistics. Appendices. Bibliography. Indexes.

1,130 citations

Journal ArticleDOI
TL;DR: In this paper, an innite system of stochastic dierential equations for the locations and weights of a collection of particles is considered, and the particles interact through their weighted empirical measure, V, and V is shown to be the unique solution of a nonlinear stochiastic partial die-rential equation (SPDE).

193 citations


"Moderate deviation principle for a ..." refers result in this paper

  • ...J. Xiong was partially supported by FDCT 076/2012/A3....

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  • ...[14] Xiong, J. (2013)....

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  • ...[11] Kurtz, T. G. and Xiong, J. (1999)....

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  • ...[7] Fatheddin, P. and Xiong, J. (2015)....

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  • ...Similar to Kurtz and Xiong [11], we can prove that − ∫...

    [...]

Journal ArticleDOI
TL;DR: In this paper, a variational representation for functionals of Brownian motion is used to avoid large deviations analysis of solutions to stochastic differential equations and related processes, where the construction and justification of the approximations can be onerous.
Abstract: 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.

183 citations