Summary.
By coupling two arbitrary riemannian connections Γ and Γ˜ on a riemannian manifold M, we perform the stochastic calculus of ɛ-variation on the path space P(M) of the manifold M. The method uses direct calculations on Ito’s stochastic differential equations. In this context, we obtain intertwinning formulas with the Ito map for first order operators on the path space P(M) of M. By a judicious choice of the second connection Γ˜ in terms of the connection Γ, we can prolongate the intertwinning 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.
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Received: 18 April 1995 / In revised form: 18 March 1996
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Airault, H., Malliavin, P. Integration by parts formulas and dilatation vector fields on elliptic probability spaces. Probab Theory Relat Fields 106, 447–494 (1996). https://doi.org/10.1007/s004400050072
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DOI: https://doi.org/10.1007/s004400050072