A remark on triviality for the two-dimensional stochastic nonlinear wave equation

Tadahiro Oh, Mamoru Okamoto, Tristan Robert

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the two-dimensional stochastic damped nonlinear wave equation (SdNLW) with the cubic nonlinearity, forced by a space-time white noise. In particular, we investigate the limiting behavior of solutions to SdNLW with regularized noises and establish triviality results in the spirit of the work by Hairer, Ryser, and Weber (2012). More precisely, without renormalization of the nonlinearity, we establish the following two limiting behaviors; (i) in the strong noise regime, we show that solutions to SdNLW with regularized noises tend to 0 as the regularization is removed and (ii) in the weak noise regime, we show that solutions to SdNLW with regularized noises converge to a solution to a deterministic damped nonlinear wave equation with an additional mass term.
Original languageEnglish
Pages (from-to)5838-5864
Number of pages27
JournalStochastic processes and their applications
Volume130
Issue number9
Early online date1 Jun 2020
DOIs
Publication statusPublished - 30 Sep 2020

Keywords

  • nonlinear wave equation
  • stochastic nonlinear wave equation
  • renormalization
  • triviality

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