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Gaussian Upper Density Estimates for Spatially Homogeneous SPDEs

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Malliavin Calculus and Stochastic Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 34))

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Abstract

We consider a general class of SPDEs in ℝ d driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring that the density of the corresponding mild solution admits an upper estimate of Gaussian type. The proof is based on the formula for the density arising from the integration-by-parts formula of the Malliavin calculus. Our result applies to the stochastic heat equation with any space dimension and the stochastic wave equation with d ∈ { 1, 2, 3}. In these particular cases, the condition on the spectral measure turns out to be optimal.

MSC Subject Classifications: 60H07, 60H15

Received 12/10/2011; Accepted 5/28/2012; Final 6/10/2012

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Acknowledgements

Part of this work has been done while the author visited the Centre Interfacultaire Bernoulli at the École Polytechnique Fédérale de Lausanne, to which he would like to thank for the financial support, as well as to the National Science Foundation. Lluís Quer-Sardanyons was supported by the grant MICINN-FEDER Ref. MTM2009-08869.

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  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Lluís Quer-Sardanyons

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  1. Lluís Quer-Sardanyons
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Correspondence to Lluís Quer-Sardanyons .

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Editors and Affiliations

  1. , Department of Statistics, Purdue University, North University Street 150, West Lafayette, 47907, Indiana, USA

    Frederi Viens

  2. , Department of Mathematics, University of Kansas, Jayhawk Blvd 1460, Lawrence, 66045, Kansas, USA

    Jin Feng

  3. Dept. Mathematics, University of Kansas, Snow Hall 405, Lawrence, 66045-2142, Kansas, USA

    Yaozhong Hu

  4. , Department of Economics and Business, University Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona, 08005, Spain

    Eulalia Nualart 

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Quer-Sardanyons, L. (2013). Gaussian Upper Density Estimates for Spatially Homogeneous SPDEs. In: Viens, F., Feng, J., Hu, Y., Nualart , E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5906-4_13

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