Convergence in Wasserstein Distance for Empirical Measures of Semilinear SPDEs

TL;DR

This paper investigates how quickly empirical measures of symmetric semilinear SPDEs converge in Wasserstein distance, revealing a logarithmic convergence rate influenced by the eigenvalues of the linear operator.

Contribution

It establishes the convergence rate in Wasserstein distance for empirical measures of semilinear SPDEs, highlighting a logarithmic order unlike the algebraic rate in finite dimensions.

Findings

Convergence in Wasserstein distance is of log order for these SPDEs.

The convergence rate depends on the eigenvalues of the linear operator.

Finite-dimensional cases exhibit algebraic convergence, contrasting with the infinite-dimensional setting.

Abstract

The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.