Gaussian Upper Density Estimates for Spatially Homogeneous SPDEs

  • Lluís Quer-SardanyonsEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)


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.


Stochastic partial differential equation Spatially homogeneous Gaussian noise Malliavin calculus 



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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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain

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