Small time expansion for a strictly hypoelliptic kernel

TL;DR

This paper derives a detailed small-time asymptotic expansion for the kernel of a hypoelliptic diffusion with non-polynomial coefficients, revealing geometric scale changes and large deviation behaviors.

Contribution

It extends small-time kernel asymptotics to strictly hypoelliptic diffusions beyond sub-elliptic or polynomial cases using a Duhamel comparison approach.

Findings

Full asymptotic expansion for small-time kernel

Identification of geometric scaling for non-trivial limits

Different scale needed for large deviation analysis

Abstract

We consider the kernel of a hypoelliptic diffusion beyond the case of sub-ellipticity or polynomial coefficients. We get a full asymptotic expansion for small times, based on a Duhamel-type comparison with an approximate polynomial kernel. As in the sub-elliptic case, some change of scale based on the geometry of some Lie brackets yields a non-trivial limit for the kernel as time goes to zero. Remarkably, a different scale is needed to observe a non-trivial large deviation principle.