Some family of q-vector fields on path spaces

TL;DR

This paper explores a class of vector fields on path spaces that facilitate an L2 Hodge-deRham theory, aiming to understand the topology of non-smooth path spaces through differential forms.

Contribution

It introduces a new class of vector fields with well-behaved brackets on path spaces, enabling the development of an L2 Hodge theory in this context.

Findings

Identifies vector fields with calculable brackets suitable for Hodge theory

Establishes a framework for differential forms on path spaces

Connects stochastic processes with geometric topology

Abstract

A great open problem is: can one learn the topology of the non-smooth path spaces with an L2 Hodge-deRham theory This one hopes to establish through a suitable complex of differential forms. Since the space is a Banach manifolds, and the Hodge theory is based on Hilbert spaces, it is trick to find such a complex. The relatively simpler Bismut tangent spaces and their tensor products, whose dual spaces are natural candidates for the complex, cannot be used, because these spaces are not necessarily closed under the Lie bracket operation if there is the effect of curvature. In this article we seek out a class of nice vector fields whose brackets behaves are calculable and behave nicely the damped tensor vector fields, using an It\^o mp, which is also used by the authors in `Special It\^o maps and an L2 Hodge theory for one forms on path spaces. Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), 145-162, CMS Conf. Proc., 28, Amer. Math. Soc., Providence, RI, 2000').