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

2026/04/16 by Charles-Edouard Bréhier, Ziyi Lei
#math.PR #math.AP

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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.

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