On the mild It\^o formula in Banach spaces

TL;DR

This paper extends the mild Itô formula to UMD Banach spaces, enabling better analysis of SPDE solutions and numerical methods, especially for establishing sharp weak convergence rates.

Contribution

The paper generalizes the mild Itô formula from Hilbert spaces to UMD Banach spaces, broadening its applicability to a wider class of SPDEs and their numerical approximations.

Findings

Generalized mild Itô formula for UMD Banach spaces

Enhanced tools for analyzing SPDE solutions

Improved weak convergence rate proofs for numerical schemes

Abstract

The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \& R\"ockner, M., A mild Ito formula for SPDEs, arXiv:1009.3526 (2012), To appear in the Trans.\ Amer.\ Math.\ Soc.] has turned out to be a useful instrument to study solutions and numerical approximations of stochastic partial differential equations (SPDEs) which are formulated as stochastic evolution equations (SEEs) on Hilbert spaces. In this article we generalize this mild It\^o formula so that it is applicable to solutions and numerical approximations of SPDEs which are formulated as SEEs on UMD (unconditional martingale differences) Banach spaces. This generalization is especially useful for proving essentially sharp weak convergence rates for numerical approximations of SPDEs.