Small-time fluctuations for the bridge in a model class of hypoelliptic diffusions of weak H\"ormander type

TL;DR

This paper investigates the small-time behavior of hypoelliptic diffusion bridges with linear drift and constant diffusivity, revealing rescaled fluctuations that converge weakly despite potential blow-up phenomena.

Contribution

It provides a detailed analysis of small-time asymptotics for a class of hypoelliptic diffusions, including explicit descriptions of fluctuation limits and examples involving Kolmogorov diffusions.

Findings

Rescaled fluctuations converge weakly in small time.

Explicit formulas for the limit fluctuation process.

Application to Kolmogorov diffusion bridges.

Abstract

We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak H\"ormander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from 0 to 0 in time 1 of an iterated Kolmogorov diffusion.