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.

翻译：迄今为止，大多数模拟条件扩散过程的方法仅限于欧几里得空间。条件过程可通过一种称为Doob $h$-变换的测度变换来构造。具体的条件类型取决于函数$h$，而该函数通常没有闭式解。为解决此问题，我们将引导过程的概念推广到流形$M$上，即用基于$M$上热核的函数替代$h$。我们考虑带漂移项的布朗运动情形，该运动利用$M$的标架丛构造，并约束其在时刻$T$击中点$x_T$。我们证明了条件过程与引导过程在律上的等价性，并给出了易于处理的Radon-Nikodym导数。随后，我们展示了如何在任意与$M$微分同胚的流形$N$上获得引导过程，而无需已知$N$上的热核。我们通过引导过程的数值模拟及基于离散时间观测的贝叶斯参数估计来验证结果，其中以环面和庞加莱圆盘为例进行说明。