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

TL;DR

This paper demonstrates that solutions of nonlinear damped stochastic wave equations in two dimensions approximate the solutions of stochastic quantization equations as a small parameter tends to zero, with results valid over large times.

Contribution

It establishes the Smoluchowski-Kramers approximation for singular stochastic wave equations in two dimensions, including cases with sine and polynomial nonlinearities.

Findings

Solutions converge to stochastic quantization equations as epsilon approaches zero.

Convergence holds over large times for sine nonlinearities.

Approximation holds even without global well-posedness for polynomial nonlinearities.

Abstract

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 dimensional torus, with polynomial and sine nonlinearities. We show that as $\epsilon \to 0$, the solutions to these equations converge to the solution of the corresponding two dimensional stochastic quantization equation. In the sine nonlinearity case, the convergence is proven over arbitrary large times, while in the polynomial case, we prove that this approximation result holds over arbitrary large times when the parameter $\epsilon$ goes to zero even with a lack of suitable global well-posedness theory for the corresponding wave equations.