Strong and weak rates of convergence in the Smoluchowski--Kramers approximation for stochastic partial differential equations

TL;DR

This paper analyzes the convergence rates of solutions to stochastic damped wave equations towards stochastic heat equations in the small-mass limit, depending on the noise regularity.

Contribution

It provides explicit strong and weak convergence rates for the Smoluchowski--Kramers approximation in stochastic PDEs, extending previous convergence results.

Findings

Strong and weak convergence rates depend on noise regularity.

For trace-class noise, rates are 1; for space-time white noise in 1D, rates are 1/2 (strong) and 1 (weak).

Results clarify how noise regularity influences convergence speed.

Abstract

We consider a class of stochastic damped semilinear wave equations, in the small-mass limit. It has previously been established that the solution converges to the solution of a stochastic semilinear heat equation. In this work we exhibit strong and weak rates of convergence in this Smoluchowski--Kramers approximation result. The rates depend on the regularity of the driving Wiener process. For instance, for trace-class noise the strong and weak rates of convergence are $1$, whereas for space-time white noise (in dimension $1$) the strong and weak rates of convergence are $1/2$ and $1$ respectively.