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We consider the heat equation associated with a class of second orderhypoelliptic H\"{o}rmander operators with constant second order term and lineardrift. We describe the possible small time heat kernel expansion on thediagonal giving a geometric characterization of the coefficients in terms ofthe divergence of the drift field and the curvature-like invariants of theoptimal control problem associated with the diffusion operator.

Curvature terms in small time heat kernel expansion for a model class of hypoelliptic H\"{o}rmander operators