Local densities for a class of degenerate diffusions

TL;DR

This paper investigates a class of degenerate diffusions generated by ultra-parabolic operators, establishing a local Itô formula for intrinsically differentiable functions and proving existence and regularity of local transition densities.

Contribution

It introduces a local Itô formula for functions differentiable in the intrinsic geometry and proves existence and regularity of transition densities for degenerate diffusions.

Findings

Established a local Itô formula for intrinsically differentiable functions.

Proved existence of local transition densities.

Demonstrated regularity properties of these densities.

Abstract

We study a class of R^d-valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Ito formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hormander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.