A version of H\"ormander's theorem in 2-smooth Banach spaces

TL;DR

This paper extends H"ormander's theorem to stochastic evolution equations in 2-smooth Banach spaces, showing the absolute continuity of the solution's law under certain conditions using Malliavin calculus.

Contribution

It introduces a version of H"ormander's theorem applicable to 2-smooth Banach spaces, broadening the scope of stochastic analysis in infinite-dimensional settings.

Findings

Absolute continuity of the law under H"ormander's bracket condition

Application of Malliavin calculus in Banach space context

Extension of classical results to 2-smooth Banach spaces

Abstract

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under H\"ormander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this result, we apply methods of the Malliavin calculus.