  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.
