Gaussian estimates for the density of the non-linear tochastic heat equation in any space dimension

Eulalia Nualart, Llus Quer-Sardanyons

Research output: Contribution to journalArticleResearchpeer-review

9 Citations (Scopus)

Abstract

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the non-linear stochastic heat equation in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise. © 2011 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)418-447
JournalStochastic Processes and their Applications
Volume122
DOIs
Publication statusPublished - 1 Jan 2012

Keywords

  • Gaussian density estimates
  • Malliavin calculus
  • Spatially homogeneous Gaussian noise
  • Stochastic heat quation

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