Small-time fluctuations for sub-Riemannian diffusion loops

TL;DR

This paper investigates the small-time behavior of sub-Riemannian diffusion loops, revealing how rescaled fluctuations converge to a limiting process and identifying conditions under which the Malliavin covariance matrices are non-degenerate.

Contribution

It introduces a novel rescaling approach that reveals the limiting distribution of diffusion loops in sub-Riemannian geometry, especially when the generator is non-elliptic.

Findings

Rescaled fluctuations converge to a limiting diffusion loop.

A non-degenerate limiting Malliavin covariance matrix is identified.

Uniform non-degeneracy of diffusion Malliavin covariance matrices is established.

Abstract

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and the generator of the diffusion process is non-elliptic at that point, the deterministic Malliavin covariance matrix is always degenerate. We identify, after a suitable rescaling, another limiting Malliavin covariance matrix which is non-degenerate, and we show that, with the same scaling, the diffusion Malliavin covariance matrices are uniformly non-degenerate. We further show that the suitably rescaled fluctuations of the diffusion loop converge to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.