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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, $P\left(M\right)$ the space of paths and $L\left(M\right)$ 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 ${C}^{\infty }$-forms in Nualart-Pardoux sense. This stochastic exterior derivative leads to ${H}^{\infty }\left(P\right)$, the entire Nualart-Pardoux cohomology, ${H}^{p}\left(P\right)$, the Bismut-Nualart-Pardoux cohomology of order $p$, and ${H}^{\infty }$(flat). It is proved that ${H}^{\infty }\left(\text{flat}\right)=H\left(M\right)$.

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:
 58D15 Manifolds of mappings 55N20 Generalized homology and cohomology theories (algebraic topology) 60H05 Stochastic integrals 58J10 Differential complexes; elliptic complexes 58A10 Differential forms (global analysis)