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We study the small-time asymptotics of the gradient and Hessian of the logarithm of the heat kernel at the cut locus, giving, in principle, complete expansions for both quantities. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic interpretation in terms of the Brownian bridge. In particular, we show that the cut locus is the set of points where the Hessian blows up faster than 1/t. We also study the distributional asymptotics and use them to compute the distributional Hessian of the energy function (that is, one-half the distance function squared).

The small-time asymptotics of the heat kernel at the cut locus