Regularity of Wiener functionals under a H\"ormander type condition of order one

TL;DR

This paper investigates the local existence and smoothness of probability densities for Wiener functionals that meet a generalized H"ormander condition involving first-order Lie brackets, extending classical diffusion process results.

Contribution

It introduces a generalized H"ormander condition of order one for Wiener functionals and analyzes their density regularity and local existence.

Findings

Established local density existence under the generalized H"ormander condition

Proved regularity results for the density of Wiener functionals

Extended classical diffusion results to a broader class of Wiener functionals

Abstract

We study the local existence and regularity of the density of the law of a functional on the Wiener space which satisfies a criterion that generalizes the H\"ormander condition of order one (that is, involving the first order Lie brackets) for diffusion processes.