Optimal gaussian density estimates for a class of stochastic equations with additive noise

David Nualart, Lluís Quer-Sardanyons

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13 Citations (Scopus)

Abstract

In this note, we establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. The proof is based on the techniques of the Malliavin calculus and a density formula obtained by Nourdin and Viens. Then, the main result is applied to the mild solution of a general class of SPDEs driven by a Gaussian noise which is white in time and has a spatially homogeneous correlation. In particular, this covers the case of the stochastic heat and wave equations in d with d < 1 and d ∈ {1, 2, 3}, respectively. The upper and lower Gaussian bounds have the same form and are given in terms of the variance of the stochastic integral term in the mild form of the equation. © 2011 World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)25-34
JournalInfinite Dimensional Analysis, Quantum Probability and Related Topics
Volume14
DOIs
Publication statusPublished - 1 Mar 2011

Keywords

  • Gaussian density estimates
  • Malliavin calculus
  • spatially homogeneous Gaussian noise
  • stochastic partial differential equations

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