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

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
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