We discuss how to simulate bridge processes by conditioning a stochastic process on a manifold whose generator is a hypo-elliptic operator. This operator is, up to a drift-term, the sub-Laplacian of a bracketgenerating 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).[-]

In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by $n \geq 2$ distinct landmark points in $\mathbb{R}^d$. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimization problem which minimizes a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold $Q$ of $n$ distinct landmark points in $\mathbb{R}^d$ can be endowed with a Riemannian metric $g$ such that the above optimization problem is equivalent to the geodesic boundary value problem for $g$ on $Q$. Despite its importance for modeling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold $(Q, g)$ is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterization of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel.[-]