Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, Shang Yuan Shiu

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We consider a family of nonlinear stochastic heat equations of the form ∂tu = Lu + σ(u)W, where W denotes space- time white noise, L the generator of a symmetric Lévy process on R, and σ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure u0. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that Lf = cf'' for some c > 0, we prove that if u0 is a finite measure of compact support, then the solution is with probability one a bounded function for all times t > 0.

Original languageEnglish
Pages (from-to)136-153
Number of pages18
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume50
Issue number1
DOIs
StatePublished - Feb 2014

Keywords

  • Singular initial data
  • The stochastic heat equation

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