Diffusion bridges for stochastic Hamiltonian systems and shape evolutions

Alexis Arnaudon, Frank van der Meulen, Moritz Schauer, Stefan Sommer

Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling stochastic evolution of human organ shapes, and in geometric mechanics for modelling turbulence parts of multi-scale fluid flows. Recently introduced models involve stochastic differential equations that govern the dynamics of a diffusion process $X$ where, in applications, $X$ is only partially observed at times $0$ and $T>0$. Conditional on these observations, the challenge is to infer parameters of the dynamics of the diffusion and to reconstruct the path $(X_t,\, t\in [0,T])$. The latter problem is known as bridge simulation. We develop a general scheme for bridge simulation in the case of finite dimensional systems of shape landmarks and singular solutions in fluid dynamics. The scheme allows for subsequent statistical inference of properties of the evolution of the shapes. We show how the approach covers stochastic landmark models for which no suitable simulation method has been proposed in the literature; that it removes restrictions of earlier approaches; that it improves the handling of the nonlinearity of the configuration space leading to more effective sampling schemes; and that it allows to generalise the common inexact matching scheme to the stochastic setting.

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