Some results for second order elliptic operators having unbounded coefficients

Abstract

By proving some estimates for the derivatives of the transition semigroup relative to a suitable stochastic equation, we prove Schauder estimates for a class of second order linear differential operators with unbounded coefficients, both in the drift and in the diffusion term. We also study asymptotic behaviour of the transition semigroup and under suitable dissipative conditions we prove existence and uniqueness of invariant measures and exponentially mixing property, uniform with respect to initial datum.