Abstract: We discuss a method, going back to work of Molchanov, for determining the small-time asymptotics of the heat kernel at the cut locus of a sub-Riemannian manifold (valid away from any abnormal geodesics), and which provides a systematic approach to a collection of related questions. When the structure of minimal geodesics between two points is known, one can (generally) determine a complete asymptotic expansion for the heat kernel and its derivatives. We describe the possible expansions, in both generic (low-dimensional) cases and non-generic cases. More generally, we give upper and lower bounds, holding uniformly on compacts. Further, we discuss the asymptotics of derivatives of the logarithm of the heat kernel, in particular showing that the (possibly vanishing) leading term is given as a cumulant with respect to a probability measure on the set of minimal geodesics. We give upper bounds on these logarithmic derivatives that hold uniformly on compacts. Returning to the probability measure on minimal geodesics, we show that it also gives the law of large numbers for the associated bridge process. This talk is based on joint work with Davide Barilari, Ugo Boscain, Grégoire Charlot, and Ludovic Sacchelli.
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