Malliavin calculus and decoupling inequalities in Banach spaces

TL;DR

This paper extends Malliavin calculus and decoupling inequalities to Banach space valued random variables, providing new tools and estimates for stochastic analysis in infinite-dimensional spaces.

Contribution

It introduces a Malliavin calculus framework for Banach spaces using radonifying operators and establishes decoupling inequalities for vector-valued random variables.

Findings

Two-sided L^p-estimates for multiple stochastic integrals in Banach spaces

Boundedness of the Malliavin derivative on Wiener-Ito chaoses

New proof of decoupling for Gaussian chaoses in UMD Banach spaces

Abstract

We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.