Intertwined diffusions

David Elworthy, University of Warwick

This is a report on joint work with Yves LeJan (Orsay) & Xue-Mei Li (Loughborough). We start by considering diffusion operators A and B on smooth manifolds M and N. These are second order semi-elliptic operators with smooth coefficients and no zero order terms. We suppose they are intertwined by a smooth surjective map from N to M . Under a coherence condition on A we obtain a decomposition of B which, given more structure, corresponds to a skew- product decomposition of the associated diffusion processes. In general it leads to a representation of the conditioned B-diffusion, conditioned on its projection to M. The basic tool is a (non-linear) "semi-connection" which the intertwining induces. There is particular geometric interest when M is compact and N is its diffeomorphism group considered as a bundle over M with projection the evaluation at a fixed point of M. The construction then relates to Narasimhan & Ramanan's construction of universal connections and the universal classifying space for the gauge group of the tangent bundle of a Riemannian manifold.