  Analogous to the characterisation of Brownian motion on a Riemannian manifold
as the development of Brownian motion on a Euclidean space, we construct
sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by
developing a canonical stochastic process arising as the lift of Brownian
motion to an associated model space. The notion of stochastic development we
introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan
connections, which take the place of the Levi-Civita connection in Riemannian
geometry. We first derive a general expression for the generator of the
stochastic process which is the stochastic development with respect to a Cartan
connection of the lift of Brownian motion to the model space. We further
provide a necessary and sufficient condition for the existence of a Cartan
connection which develops the canonical stochastic process to the
sub-Riemannian diffusion associated with the sub-Laplacian defined with respect
to the Popp volume. We illustrate the construction of a suitable Cartan
connection for free sub-Riemannian structures with two generators and we
discuss an example where the condition is not satisfied.
