The path space analysis using “redundant noise” carried out jointly with Yves Le Jan & Xue-Mei Li covered the case of a substantial class of degenerate diffusion measures. I will review this and show how it can lead to Bismut formulae given strong Hormander conditions, with special reference to diffusions “transverse to a Riemannian foliation with totally geodesic leaves”, a concept which will be carefully explained.

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