Stochastic cohomology of the frame bundle of the loop space

TL;DR

This paper investigates stochastic differential forms on the frame bundle of the loop space, introducing an exterior stochastic differential derivative to analyze curvature phenomena under regularity assumptions.

Contribution

It develops a framework for stochastic differential forms on the loop space's frame bundle and defines an exterior stochastic differential derivative.

Findings

Defined an exterior stochastic differential derivative for forms.

Analyzed curvature phenomena via Lie brackets of horizontal vector fields.

Established regularity assumptions for the kernels of differential forms.

Abstract

We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose some regularity assumptions over the kernels of the differential forms. This allows us to define an exterior stochastic differential derivative over these forms.