The talk will concentrate on the recent progress in two different infinite-dimensional hypoelliptic settings. One is of infinite-dimensional Heisenberg groups equipped with a natural sub-Riemannian (hypoelliptic) metric, and another of the horizontal path space of a totally geodesic Riemannian foliation. In the first case the heat kernel measure satisfies a Cameron-Martin type quasi-invariance, while in the second case the horizontal Wiener measure is quasi-invariant with respect to the flows generated by suitable tangent processes. For the path space we also prove Driver's integration by parts formula. Methods are geometric, though very different in nature: generalized curvature-dimension inequalities, functional inequalities, and appropriate metric connections. A different approach for the infinite-dimensional Heisenberg group has been developed by B. Driver, N. Eldredge and T. Melcher. This talk is based on the joint work with F. Baudoin, Q. Feng, T. Melcher

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