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

Tadahiro Oh, Mamoru Okamoto and Tristan Robert

Stochastic Processes and their Applications, 2020, vol. 130, issue 9, 5838-5864

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 et al. (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.

Keywords: Nonlinear wave equation; Stochastic nonlinear wave equation; Renormalization; Triviality (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spa.2020.05.010

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