# Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise

## TL;DR

This paper establishes convergence rates for spectral Galerkin and finite element space approximations of semi-linear parabolic SPDEs with multiplicative noise, focusing on strong and pathwise convergence in a Hilbert space.

## Contribution

It provides the first detailed analysis of pathwise convergence rates for these approximations of semi-linear SPDEs with multiplicative noise.

## Key findings

- Spectral Galerkin and finite element methods achieve strong convergence in SPDE approximations.
- Convergence rates depend on the regularity of the solution and the approximation space.
- The results are applicable to a broad class of semi-linear SPDEs with multiplicative noise.

## Abstract

We provide convergence rates for space approximations of semi-linear stochastic differential equations with multiplicative noise in a Hilbert space. The space approximations we consider are spectral Galerkin and finite elements, and the type of convergence we consider is strong and almost sure uniform convergence, i.e., pathwise convergence. The proofs are based on a previously published perturbation result for such equations.