Markovian connection, curvature and Weitzenböck formula on the Riemannian path space (Q2708193)

The object of this paper is to prove a Bochner-Weitzenböck type formula on the path space \(P(M)\) of a \(d\)-dimensional Riemannian manifold \(M\). For this, the author considers the Markovian connection \(\nabla\) introduced by Cruzeiro and Malliavin, the intrinsic gradient \(D\) on path space and its associated Laplacian \(\Delta^P\). He defines a pairing \(\langle z , \xi \rangle\) between a simple vector field \(z\) on \(P(M)\) and a tangent process \(\xi\) by taking the Stratonovich integral of \(z\) with respect to \(\xi\). He then states the existence of a tangent process \(\widehat{\text{Ric}}^P k\) for all \(k\) in the Cameron-Martin space \(H\), and of an \(H\)-valued integrable random variable \(D^1(\nabla F)\), such that the Weitzenböck type formula NEWLINE\[NEWLINE\langle \nabla \Delta^P F,k\rangle_H = \langle (\Delta^P + D^1)(\nabla F),k\rangle_H - \langle \widehat{\text{Ric}}^P k,\nabla F\rangle NEWLINE\]NEWLINE holds for \(F\) a cylindrical functional on \(P(M)\) and \(k\in H\). The term \(D^1(\nabla F)\) vanishes when the torsion of \(M\) satisfies to the condition of Driver, i.e. when it is skew symmetric. The proofs of these results will appear in a forthcoming paper.