Integration by parts formulas and dilatation vector fields on elliptic probability spaces. (English) Zbl 0867.60031

Airault, Helene; Malliavin, Paul

Probab. Theory Relat. Fields 106, No. 4, 447-494 (1996).

Summary: By coupling two arbitrary Riemannian connections \(\Gamma\) and \(\widetilde\Gamma\) on a Riemannian manifold \(M\), we perform the stochastic calculus of \(\varepsilon\)-variation on the path space \(P(M)\) of the manifold \(M\). The method uses direct calculations on Itô’s stochastic differential equations. In this context, we obtain intertwinning formulas with the Itô map for first-order operators on the path space \(P(M)\) of \(M\). By a judicious choice of the second connection \(\widetilde\Gamma\) in terms of the connection \(\Gamma\), we can prolongate the interwinning formulas to second-order differential operators. Thus, we obtain expressions of heat operators on the path space \(P(M)\) of a Riemannian manifold \(M\) endowed with an arbitrary connection. The integration by parts of the Laplacians on \(P(M)\) leads us to the notion of dilatation vector field on the path space.

MSC:

60H07	Stochastic calculus of variations and the Malliavin calculus
60D05	Geometric probability and stochastic geometry
28D05	Measure-preserving transformations

Keywords:

stochastic calculus of epsilon-variation; stochastic differential equations; dilatation vector field