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Bismut-Nualart-Pardoux cohomology and entire Hochschild cohomology. (Cohomologie de Bismut-Nualart-Pardoux et cohomologie de Hochschild entière.) (French) Zbl 0870.58011
Azéma, J. (ed.) et al., Séminaire de probabilités XXX. Berlin: Springer. Lect. Notes Math. 1626, 68-99 (1996).

Let M be a compact, finite-dimensional, Riemannian manifold, P(M) the space of paths and L(M) the space of free loops on M.

The article consists of two parts. The first part, using a regularity defined by D. Nualart and E. Pardoux [Probab. Theory Relat. Fields 78, No. 4, 535-581 (1988; Zbl 0629.60061)], is building a version of stochastic exterior derivative on the space of C -forms in Nualart-Pardoux sense. This stochastic exterior derivative leads to H (P), the entire Nualart-Pardoux cohomology, H p (P), the Bismut-Nualart-Pardoux cohomology of order p, and H (flat). It is proved that H (flat)=H(M).

In the second part, following E. Getzler, J. Jones and S. Petrack [Topology 30, No. 3, 339-371 (1991; Zbl 0729.58004)] a commutative diagram of complexes is used to prove the equality between the entire Hochschild cohomology and the stochastic cohomology on the loop space.

MSC:
58D15Manifolds of mappings
55N20Generalized homology and cohomology theories (algebraic topology)
60H05Stochastic integrals
58J10Differential complexes; elliptic complexes
58A10Differential forms (global analysis)