We discuss how to simulate bridge processes by conditioning a stochastic process on a manifold whose generator is a hypoelliptic operator. This operator is, up to a drift-term, the sub-Laplacian of a bracket-generating sub-Riemannian structure, meaning in particular that it has positive smooth density everywhere. The logarithmic gradient of this density is called the score, and we show that it is needed to describe the generator of the bridge process. We therefore discuss several methods for how we can estimate the score using a neural network, with examples.

The results are from a joint work with Stefan Sommer (Copenhagen) and Karen Habermann (Warwick).