We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov–Fokker–Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small-time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous Hörmander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the Hörmander condition.

Kolmogorov–Fokker–Planck operators in dimension two: heat kernel and curvature

Barilari D.;Boarotto F.
2018

Abstract

We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov–Fokker–Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small-time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous Hörmander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the Hörmander condition.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11577/3368979
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