Vol. 15, No. 1, 2022

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Diffusion in small time in incomplete sub-Riemannian manifolds

Ismael Bailleul and James Norris

Vol. 15 (2022), No. 1, 63–84
Abstract

For incomplete sub-Riemannian manifolds and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.

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Keywords
sub-Riemannian, heat kernel, diffusion
Mathematical Subject Classification 2010
Primary: 35K08, 58J65, 60J60
Milestones
Received: 26 April 2019
Revised: 21 February 2020
Accepted: 15 September 2020
Published: 16 March 2022
Authors
Ismael Bailleul
CNRS, Institut de Recherche Mathematiques de Rennes, UMR 6625
University of Rennes 1
Rennes
France
James Norris
Statistical Laboratory
Centre for Mathematical Sciences
University of Cambridge
Cambridge
United Kingdom