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，条件扩散过程的大多数模拟方法仅限于欧几里得环境。条件过程可通过称为Doob $h$-变换的测度变化构建。特定的条件类型依赖于函数 $h$，而该函数通常无法以闭合形式明确表达。为解决此问题，我们将引导过程的概念推广到流形 $M$ 上，用基于 $M$ 上热核的函数替代 $h$。我们考虑利用 $M$ 的标架丛构造的带漂移布朗运动，其条件设定为在时间 $T$ 击中点 $x_T$。我们证明了条件过程与引导过程在具有可计算 Radon-Nikodym 导数条件下的等价性。进一步地，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，且无需假设已知 $N$ 上的热核。我们通过数值模拟及一个参数估计实例（环面上扩散过程的离散时间观测）验证了所得结果。