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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 \(C^\infty\)-forms in Nualart-Pardoux sense. This stochastic exterior derivative leads to \(H^\infty(P)\), the entire Nualart-Pardoux cohomology, \(H^p(P)\), the Bismut-Nualart-Pardoux cohomology of order \(p\), and \(H^\infty\)(flat). It is proved that \(H^\infty (\text{flat}) =H(M)\).
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.
For the entire collection see [Zbl 0840.00041].

MSC:

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