Density estimates for differential equations of second order satisfying a weak Hoermander condition

TL;DR

This paper extends Hoermander's classical hypoelliptic second order PDE results to cases with globally Lipschitz coefficients satisfying the classical condition on a dense set, ensuring the existence of densities.

Contribution

It provides an analytical extension of Hoermander's theorem for second order equations with coefficients satisfying the classical condition on a dense set, recovering classical density estimates.

Findings

Density exists under extended conditions

Classical density estimates are recovered

Analytical approach complements Malliavin calculus

Abstract

We prove an extension of Hoermander's classical result on hypoelliptic second order equations, where the coefficients of the related vector fields are globally Lipschitz and satisfy the classical Hoermander condition on a dense set while the density still exists in a classical sense. Furthermore, Hoermander's classical result and related density estimates based on Malliavin calculus are recovered from an analytical point of view.