流形上条件扩散的模拟
Simulating conditioned diffusions on manifolds
AI总结：
针对现有条件扩散模拟方法多局限于欧氏空间、条件函数h无闭式解的问题，将引导过程推广到流形，证明其与条件过程律等价，还可拓展到微分同胚流形，并通过环面和庞加莱圆盘的实验验证了方法有效性。
摘要：
迄今为止，大多数模拟条件扩散的方法都局限于欧几里得情形。条件过程可以通过一种被称为Doob h变换的测度变换来构建。具体的条件类型取决于函数h，而该函数通常没有闭式解。为解决这一问题，我们将引导过程的概念推广到流形M上，即用基于M上热核的函数来替代h。我们考虑了通过M的标架丛构建的、带漂移的布朗运动在时刻T击中x_T点的条件情形。我们证明了条件过程与引导过程的律等价，且具有可计算的Radon-Nikodym导数。随后，我们展示了如何在无需知晓N上热核的情况下，在任意与M微分同胚的流形N上得到引导过程。我们通过引导过程的数值模拟以及基于离散时间观测的贝叶斯参数估计验证了我们的结果，为此我们同时考虑了环面（torus）和庞加莱圆盘（Poincaré disk）。
英文摘要：
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 Poincaré disk.
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