  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.
