Stochastic processes on curved spaces

摘要：随机过程在(平坦的)欧氏空间上是很好理解的。但最近有一些有趣的过程，其状态空间是在任意(曲线)流形上的。半鞅的概念在流形上是定义良好的，只要流形上给出了黎曼度量，或者更广泛地说只是一个连通，流形上的半鞅和欧氏空间上的半鞅之间实际上存在着简洁的联系。与此相关的问题是试图理解流形上过程的结构，并看看如何模拟它们。我们将首先关注最简单的弯曲流形--任意维单位球面--及其上的正则过程--布朗运动。在这种特殊情况下，球的对称性和布朗运动在这些对称性下的不变性对理解这一过程起着关键作用。我们希望利用这一事实来获得这一进程的结构性后果。特别令人感兴趣的是所谓的歪积分解，它将一个过程分解成两个维度较低且行为良好的过程，通常发生在给定的度量不是乘积度量而是所谓的翘曲乘积度量的乘积空间上。利用这种分解，人们在理论上可以将过程简化为一系列相关的一维过程，这是众所周知的。特别地，人们通常可以模拟一维过程，而改变分解可以产生对原始的更多维过程有用的模拟算法。另一个我们想要解决的问题是如何在一般流形上以规范和合适的方式定义一类过程-Levy过程。Levy过程最初定义在欧几里得空间上，一个特别有趣的特征是它们本质上是表现出跳跃的最简单过程类。虽然大多数理论都转化为李群设置(特别是欧几里德空间也是李群)，但关于在一般流形上可以做些什么，我们知道的要少得多。其中一个问题是，增量的概念在一般流形上没有意义，加上跳跃是另一个问题，因为理论上跳跃可以把我们带到流形上的任何地方，而且流形通常只在局部表现良好，而全局结构可能非常复杂。关于流形上的一些Levy过程的定义已经做了一些工作，但似乎有一些需要改进的地方，因为似乎存在更多的过程，这些过程理所当然地被称为Levy过程，并且没有包括在以前的构造中。我们的目标将是使用微分几何的某些工具-主纤维丛、联络、框架丛、(反)发展-来解决这个问题，并试图对流形上可以被正确地称为Levy过程的最大类过程进行分类，当我们考虑欧几里德空间和Lie群时，这个概念应该产生经典的Levy过程的概念，另外，我们应该在这种设置下得到所有可能的Levy过程(对于当前的构造来说不是这样)。

Summary: Stochastic processes are well understood on (flat) Euclidean spaces. But recently there are of interest processes whose state space is on arbitrary (curved) manifold. The notion of semimartingale is well-defined on a manifold and there is actually a succinct connection between those semimartingales on manifolds and semimartingales on Euclidean spaces as long as the manifold has given Riemannian metric or more generally just a connection.Related problem to above is trying to understand structure of processes on manifolds and seeing how they can be simulated. We will foremost focus on the simplest curved manifold - unit sphere in arbitrary dimension - and on canonical process on it - Brownian motion. It this particular case symmetries of the sphere and additionally invariance of Brownian motion under those symmetries play key role to understanding the process. We wish to utilise this fact to obtain structural consequences for the process. Of particular interest would be so called skew-product decomposition which decomposes a process into two less dimensional and well behaved processes and usually occurs on product spaces where the metric given is not a product one, but so called warped-product one. Using this decomposition one could in theory reduce the process to a series of related one-dimensional processes which are well understood. In particular one can usually simulate one-dimensional processes and turning the decomposition around could yield useful simulation algorithms for the original more dimensional process.Another problem we wish to tackle is to how to define in a canonical and suitable way a certain class of processes - Levy processes - on general manifolds. Levy processes are originally defined on Euclidean spaces and one particularly interesting feature is that they are essentially simplest class of processes exhibiting jumps. While most of the theory translates to a Lie group setup (Euclidean spaces are in particular also Lie groups), much less is known on what can be done on a general manifold. One of the problems is that a notion of increment does not make sense on a general manifold and addition of jumps is the other problem, since jumps could theoretically take us anywhere on the manifold and manifold are usually well behaved only locally, whereas global structure can be very intricate. There has been some work done on defining some Levy processes on manifolds, but it seems that there are improvements to be made since there seem to exists more processes which would rightfully be dubbed Levy processes and were not included in previous constructions. Our goal will be to use certain tools from differential geometry - principal fibre bundles, connection, frame bundles, (anti-)development - to tackle this problem and try to classify maximal class of processes which could be rightfully called Levy processes on manifolds and this notion should yield classical notion of Levy processes when we consider Euclidean spaces and Lie groups and additionally we should get all possible Levy processes in this setup (which is not the case for current constructions).