Super-convergence analysis on exponential integrator for stochastic heat equation driven by additive fractional Brownian motion

TL;DR

This paper establishes a super-convergence result for an exponential integrator applied to the stochastic heat equation driven by additive fractional Brownian motion with Hurst parameter greater than 0.5, demonstrating strong order one accuracy.

Contribution

It presents the first super-convergence result in the temporal direction for full discretizations of SPDEs driven by infinite-dimensional fractional Brownian motion.

Findings

Proves strong order one convergence of the exponential integrator.

First super-convergence result for SPDEs with fractional Brownian motion.

Utilizes Malliavin calculus and smoothing effects in the proof.

Abstract

In this paper, we consider the strong convergence order of the exponential integrator for the stochastic heat equation driven by an additive fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$. By showing the strong order one of accuracy of the exponential integrator under appropriote assumptions, we present the first super-convergence result in temporal direction on full discretizations for stochastic partial differential equations driven by infinite dimensional fractional Brownian motions with Hurst parameter $H\in(\frac12,1)$. The proof is a combination of Malliavin calculus, the $L^p(\Omega)$-estimate of the Skorohod integral and the smoothing effect of the Laplacian operator.