Title | Volume variation and heat kernel for affine control problems |

Publication Type | Thesis |

Year of Publication | 2015 |

Authors | Paoli, E |

University | SISSA |

Keywords | Heat kernel asymptotics |

Abstract | 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. |

Custom 1 | 35290 |

Custom 2 | Mathematics |

Custom 4 | -1 |

Custom 5 | MAT/05 |

Custom 6 | Submitted by epaoli@sissa.it (epaoli@sissa.it) on 2015-11-26T08:47:31Z |

## Volume variation and heat kernel for affine control problems

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