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We study a family of nonlinear damped wave equations indexed by a parameter$\epsilon >0$ and forced by a space-time white noise on the two dimensionaltorus, with polynomial and sine nonlinearities. We show that as $\epsilon \to0$, the solutions to these equations converge to the solution of thecorresponding two dimensional stochastic quantization equation. In the sinenonlinearity case, the convergence is proven over arbitrary large times, whilein the polynomial case, we prove that this approximation result holds overarbitrary large times when the parameter $\epsilon$ goes to zero even with alack of suitable global well-posedness theory for the corresponding waveequations.

Smoluchowski-Kramers approximation for the singular stochastic wave equations in two dimensions