TY - JOUR T1 - Volume geodesic distortion and Ricci curvature for Hamiltonian dynamics JF - arXiv preprint arXiv:1602.08745 Y1 - 2016 A1 - Andrei A. Agrachev A1 - Davide Barilari A1 - Elisa Paoli ER - TY - THES T1 - Volume variation and heat kernel for affine control problems Y1 - 2015 A1 - Elisa Paoli KW - Heat kernel asymptotics AB - In 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. PB - SISSA U1 - 35290 U2 - Mathematics U4 - -1 U5 - MAT/05 ER -