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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.

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