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A Laplace Principle for a Stochastic Wave Equation in Spatial Dimension Three

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Stochastic Analysis 2010

Abstract

We consider a stochastic wave equation in spatial dimension three, driven by a Gaussian noise, white in time and with a stationary spatial covariance. The free terms are non-linear with Lipschitz continuous coefficients. Under suitable conditions on the covariance measure, Dalang and Sanz-Solé (“Memoirs of the AMS, 199, 931, 2009”) have proved the existence of a random field solution with Hölder continuous sample paths, jointly in both arguments, time and space. By perturbing the driving noise with a multiplicative parameter ε ∈ ]0, 1], a family of probability laws corresponding to the respective solutions to the equation is obtained. Using the weak convergence approach to large deviations developed in (“Dupuis and Ellis, A weak convergence approach to the theory of large deviations, Wiley, 1997”), we prove that this family satisfies a Laplace principle in the Hölder norm.

MSC (2010): 60H15, 60F10

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Acknowledgements

Supported by the grant MTM 2009-07203 from the Dirección General de Investigación, Ministerio de Ciencia e Innovación, Spain.

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Authors and Affiliations

  1. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Cortes Catalanes 585, 08007, Barcelona, Spain

    Víctor Ortiz-López & Marta Sanz-Solé

Authors
  1. Víctor Ortiz-López
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  2. Marta Sanz-Solé
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Correspondence to Marta Sanz-Solé .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

    Dan Crisan

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Ortiz-López, V., Sanz-Solé, M. (2011). A Laplace Principle for a Stochastic Wave Equation in Spatial Dimension Three. In: Crisan, D. (eds) Stochastic Analysis 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15358-7_3

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