We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We completely describe the small time heat kernel expansions on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the drift field and the curvature-like invariants of the optimal control problem associated with the diffusion operator.

10aCurvature10aHypoelliptic heat equation10aSmall time asymptotics1 aBarilari, Davide1 aPaoli, Elisa uhttp://www.sciencedirect.com/science/article/pii/S0362546X1730229801212nas a2200145 4500008004100000022001400041245008800055210007000143260000800213300001400221490000700235520076100242100001701003856004601020 2017 eng d a1573-869800aSmall Time Asymptotics on the Diagonal for Hörmander's Type Hypoelliptic Operators0 aSmall Time Asymptotics on the Diagonal for Hörmanders Type Hypoe cJan a111–1430 v233 aWe compute the small time asymptotics of the fundamental solution of Hörmander's type hypoelliptic operators with drift, on the diagonal at a point x0. We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x0, then so is also the original control problem, and in this case we show that the fundamental solution blows up as t−N/2\$\backslashphantom {\backslashdot {i}\backslash!}t^{-\backslashmathcal {N}/2}\$, where N\$\backslashphantom {\backslashdot {i}\backslash!}\backslashmathcal {N}\$is a number determined by the Lie algebra at x0 of the fields, that define the hypoelliptic operator.

1 aPaoli, Elisa uhttps://doi.org/10.1007/s10883-016-9321-z00469nas a2200109 4500008004100000245007600041210006900117100002500186700002100211700001700232856011000249 2016 eng d00aVolume geodesic distortion and Ricci curvature for Hamiltonian dynamics0 aVolume geodesic distortion and Ricci curvature for Hamiltonian d1 aAgrachev, Andrei, A.1 aBarilari, Davide1 aPaoli, Elisa uhttps://www.math.sissa.it/publication/volume-geodesic-distortion-and-ricci-curvature-hamiltonian-dynamics02046nas a2200121 4500008004100000245006500041210006500106260001000171520159900181653002801780100001701808856009901825 2015 en d00aVolume variation and heat kernel for affine control problems0 aVolume variation and heat kernel for affine control problems bSISSA3 aIn this thesis we study two main problems. The first one is the small-time heat kernel expansion on the diagonal for second order hypoelliptic opeartors. We consider operators that can depend on a drift field and that satisfy only the weak Hörmander condition. In a first work we use perturbation techniques to determine the exact order of decay of the heat kernel, that depends on the Lie algebra generated by the fields involved in the hypoelliptic operator. We generalize in particular some results already obtained in the sub-Riemannian setting. In a second work we consider a model class of hypoelliptic operators and we characterize geometrically all the coefficients in the on-the diagonal asymptotics at the equilibrium points of the drift field. The class of operators that we consider contains the linear hypoelliptic operators with constant second order part on the Euclidean space. We describe the coefficients in terms only of the divergence of the drift field and of curvature-like invariants, related to the minimal cost of geodesics of the associated optimal control problem. In the second part of the thesis we consider the variation of a smooth volume along a geodesic. The structure of the manifold is induced by a quadratic Hamiltonian and the geodesic in described as the projection of the Hamiltonian flow. We find an expansion similar to the classical Riemannian one. It depends on the curvature operator associated to the Hamiltonian, on the symbol of the geodesic and on a new metric-measure invariant determined by the symbol of the geodesic and by the given volume.10aHeat kernel asymptotics1 aPaoli, Elisa uhttps://www.math.sissa.it/publication/volume-variation-and-heat-kernel-affine-control-problems