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Pré-Publication, Document De Travail Année : 2012

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

Davide Barilari
Ugo Boscain

Résumé

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 $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, 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 we get the expansion $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.

Dates et versions

hal-00687651 , version 1 (13-04-2012)

Identifiants

Citer

Davide Barilari, Ugo Boscain, Robert W. Neel. Small time heat kernel asymptotics at the sub-Riemannian cut locus. 2012. ⟨hal-00687651⟩
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