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

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, when the endpoints are joined by a unique path of minimal energy, the conditioned diffusion converges weakly to that path. We show further that, when the endpoints lie outside the sub-Riemannian cut locus, 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 minimal second variation of the energy functional at the minimal path, the formulation of which is new in this context.