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

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.