Small-time fluctuations for the bridge of a sub-Riemannian diffusion

TL;DR

This paper studies small-time behavior of conditioned sub-Riemannian diffusions, showing that fluctuations around the minimal energy path converge to a Gaussian limit characterized by the bicharacteristic flow and a novel second variation of energy.

Contribution

It introduces a new formulation of the second variation of the energy functional for sub-Riemannian diffusions and characterizes Gaussian fluctuations in this setting.

Findings

Fluctuations converge to a Gaussian limit outside the cut locus.

The Gaussian limit is characterized by bicharacteristic flow.

A new second variation of the energy functional is formulated.

Abstract

We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.