  For incomplete sub-Riemannian manifolds, and for an associated second-order
hypoelliptic operator, which need not be symmetric, we identify two alternative
conditions for the validity of Gaussian-type upper bounds on heat kernels and
transition probabilities, with optimal constant in the exponent. Under similar
conditions, we obtain the small-time logarithmic asymptotics of the heat
kernel, and show concentration of diffusion bridge measures near a path of
minimal energy. The first condition requires that we consider points whose
distance apart is no greater than the sum of their distances to infinity. The
second condition requires only that the operator not be too asymmetric.
