Exact Small Time Equivalent for the density of the Circular Langevin Diffusion

TL;DR

This paper derives an exact small-time density approximation for the circular Langevin diffusion, a hypoelliptic non-Gaussian process, including complex singular cases related to conjugate points.

Contribution

It provides the first explicit small-time density equivalent for the circular Langevin diffusion, addressing both regular and singular cases in hypoelliptic settings.

Findings

Explicit small-time density equivalent derived

Handles complex singular cases related to conjugate points

Advances understanding of hypoelliptic diffusion processes

Abstract

A small time equivalent of the density is obtained for the circular analogue of the Langevin diffusion, which is strictly hypoelliptic (and non-Gaussian), hence of a different nature as the known sub-Riemannian case. The singular case, analogous to the case of conjugate points (the cut-locus problem) in the sub-Riemannian framework, is totally handled too, though much more difficultly.