Simulating conditioned diffusions on manifolds

To date, most methods for simulating conditioned diffusions are limited to the Euclidean setting. The conditioned process can be constructed using a change of measure known as Doob’s ℎ-transform. The specific type of conditioning depends on a function ℎ which is typically unknown in closed form. To resolve this, we extend the notion of guided processes to a manifold M, where one replaces ℎ by a function based on the heat kernel on M. We consider the case of a Brownian motion with drift, constructed using the frame bundle of M, conditioned to hit a point T_T at time T. We prove equivalence of the laws of the conditioned process and the guided process with a tractable Radon-Nikodym derivative. Subsequently, we show how one can obtain guided processes on any manifold N that is diffeomorphic to M without assuming knowledge of the heat kernel on N. We illustrate our results with numerical simulations of guided processes and Bayesian parameter estimation based on discrete-time observations. For this, we consider both the torus and the Poincaré disk.