Stochastic Diffeology and Homotopy

Abstract

There are two stochastic cohomology theories of the loop space: In the first, we endow the loop space with the Brownian bridge measure if the underlying manifold is Riemannian. Sobolev spaces of forms are defined because a Hilbert tangent space is used, although the Brownian loop is only continuous ([8], [2]). To a random form is associated a series of numbers, that is its Sobolev norms, and the stochastic exterior derivative acts continuously over the intersection of Sobolev spaces ([10], [11], [13]). It is shown that if the manifold is simply connected, then the Sobolev stochastic cohomology of forms in this sense (a metric invariant) is equal to the Hochschild cohomology of forms over the manifold ([11], [13]) and therefore to the de Rham cohomology of the smooth loop space (a differentiable invariant). This result is a stochastic generalization of a result of Adams ([1]) and Chen ([4]), which says that the Hochschild cohomology is equal to the cohomology of smooth loops.