*Preprint*

**Inserted:** 6 sep 2017

**Last Updated:** 25 sep 2017

**Year:** 2017

**Notes:**

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 ̈ormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the H ̈ormander condition.

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