Calculus of Variations and Geometric Measure Theory

D. Barilari - F. Boarotto

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

created by barilari on 06 Sep 2017
modified on 16 Apr 2021


Published Paper

Inserted: 6 sep 2017
Last Updated: 16 apr 2021

Journal: Journal of Evolution Equations
Year: 2018


We consider the heat equation associated with a class of hypoelliptic operators of Fokker-Planck-Kolmogorov 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 associated with non-homogeneous H ╠łormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the H ╠łormander condition.