Dissipation in Parabolic SPDEs

Davar Khoshnevisan, Kunwoo Kim, Carl Mueller, Shang Yuan Shiu

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability.

Original languageEnglish
Pages (from-to)502-534
Number of pages33
JournalJournal of Statistical Physics
Volume179
Issue number2
DOIs
StatePublished - 1 Apr 2020

Keywords

  • Dissipation
  • Intermittency
  • Stochastic partial differential equations
  • White noise

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