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

## TL;DR

This paper studies the two-dimensional stochastic damped nonlinear wave equation with cubic nonlinearity and white noise, revealing that solutions trivialize or converge to deterministic equations depending on noise strength, without requiring renormalization.

## Contribution

It establishes new triviality and convergence results for the 2D stochastic nonlinear wave equation with regularized noise, without renormalization, in both strong and weak noise regimes.

## Key findings

- Solutions tend to zero in the strong noise regime.
- Solutions converge to a deterministic wave equation in the weak noise regime.
- No renormalization needed for these limiting behaviors.

## 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.