Intermittency for the Hyperbolic Anderson Model with rough noise in space

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Abstract

© 2016 Elsevier B.V. In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index H∈([Formula presented], [Formula presented]). Initial data are assumed to be constant. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the p-th moment the solution, for any p≥2. Condition H>[Formula presented] turns out to be necessary for the existence of solution. Secondly, we show that this solution coincides with the one obtained by the authors in a recent publication, in which the solution is interpreted in the Itô sense. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent.
Original languageEnglish
Pages (from-to)2316-2338
JournalStochastic Processes and their Applications
Volume127
Issue number7
DOIs
Publication statusPublished - 1 Jul 2017

Keywords

  • Intermittency
  • Malliavin calculus
  • Stochastic partial differential equations
  • Stochastic wave equation

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