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

Tadahiro Oh (1) , Mamoru Okamoto , Tristan Robert (2)

1 Edin. - University of Edinburgh
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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.

Mots clés: Nonlinear wave equation; Stochastic nonlinear wave equation; Renormalization; Triviality

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Tadahiro Oh, Mamoru Okamoto, Tristan Robert. A remark on triviality for the two-dimensional stochastic nonlinear wave equation. Stochastic Processes and their Applications, 2020, 130 (9), pp.5838-5864. ⟨10.1016/j.spa.2020.05.010⟩. ⟨hal-02434795⟩