HARNACK INEQUALITIES FOR HYPOELLIPTIC EVOLUTION OPERATORS: GEOMETRIC ISSUES AND APPLICATIONS

We consider linear second order Partial Differentia Equations in the form of "sum of squares of Hormander's vector fields plus a drift term" on a given domain. We prove that a Harnack inequality holds for every compact subset of the interior of the attainable set defined in terms of the vector fields considered. We then apply the Harnack's inequalities to prove asymptotic lower bounds of the joint density of a wide class of stochastic processes. Analogous upper bounds for the density are proved by Mallilavin's calculus.