Simulating conditioned diffusions on manifolds

TL;DR

This paper extends methods for simulating conditioned diffusions from Euclidean spaces to manifolds, using heat kernels and guided processes, with applications demonstrated on the torus and Poincaré disk.

Contribution

It introduces a novel approach to construct guided processes on manifolds using heat kernels, overcoming the need for explicit heat kernel knowledge on the target manifold.

Findings

Proves equivalence of conditioned and guided process laws on manifolds.

Develops methods to construct guided processes on diffeomorphic manifolds without heat kernel.

Demonstrates numerical simulations and Bayesian inference on complex manifolds.

Abstract

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 $x_T$ 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 Poincar\'e disk.