Explicit parametrix and local limit theorems for some degenerate diffusion processes

TL;DR

This paper develops explicit Gaussian controls and a local limit theorem for a class of rank 2 degenerate diffusion processes, leveraging Gaussian approximation techniques to handle the singularities caused by degeneracy.

Contribution

It introduces a parametrix representation for the density of degenerate diffusions and establishes a local limit theorem for Markov chain approximations, extending Gaussian approximation methods.

Findings

Derived explicit Gaussian controls for degenerate diffusions

Established a local limit theorem with standard convergence rate

Applied Gaussian approximation techniques to degenerate processes

Abstract

For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of the density from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. We then give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the "weak" degeneracy allows to exploit the techniques first introduced by Konakov and Molchanov and then developed by Konakov and Mammen that rely on Gaussian approximations.