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Stochastic Adams theorem for a general compact manifold. (English) Zbl 1037.58025

Let M be a compact Riemannian simply connected manifold, and let L x (M) be the loop space based at xM, endowed with the Brownian bridge law and with H-derivation.

The main result here is that over L x (M) the stochastic cohomology of smooth forms equals the Hochschild cohomology. This is the stochastic analysis counterpart of the analogous theorem by Adams, which concerns smooth loops.

The main step in the long proof of this result is to identify the cohomologies of smooth forms over L x (M) and over the space of paths on M which start from x and arrive in some open neighborhood of x.


MSC:
58J65Diffusion processes and stochastic analysis on manifolds
60H07Stochastic calculus of variations and the Malliavin calculus
60J65Brownian motion