Small-time heat kernel asymptotics at the sub-riemannian cut locus

Abstract

For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from x with roughly “how much” y is conjugate to x. This is done under the hypothesis that all minimizers connecting x to y are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre 4t log pt(x, y) → −d2(x, y) for t → 0, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion pt(x, y) ∼ t−5/4 exp(−d2(x, y)/4t) where y is reached from a - Riemannian point x by a minimizing geodesic which is conjugate at y. © 2012 Journal of Differential Geometry. © 2012 Applied Probability Trust.