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 and an example of parameter estimation where a diffusion process on the torus is observed discretely in time.

翻译：迄今为止，大多数模拟条件扩散的方法仅限于欧几里得空间。条件过程可以通过一种称为Doob h-变换的测度变化来构造，其特定条件化类型依赖于通常无法以闭式表达的h函数。为解决这一问题，我们将引导过程的概念推广至流形M，用基于M上热核的函数替代h函数。我们考虑利用M的标架丛构造的带漂移布朗运动，该过程条件化于在时间T到达目标点$x_T$。我们证明了条件过程与引导过程的测度等价性，并得到了可解析计算的Radon-Nikodym导数。进而，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，而无需假设N上的热核已知。最后通过数值模拟和环面上扩散过程离散时间观测的参数估计实例验证了我们的结论。