Harnack inequalities for hypoelliptic evolution operators: geometric issues and applications

Abstract

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

Keywords

• hypoelliptic equations, Harnack inequality
• potential theory
• Malliavin calculus