Stochastic processes on curved spaces

摘要：随机过程在（平坦）欧几里得空间上被很好地理解。但近年来出现了一些状态空间在任意（弯曲）流形上的过程。流形上的半鞅的概念是有定义的实际上流形上的半鞅和欧几里德空间上的半鞅之间有一个简洁的联系只要流形给出黎曼度规或者更一般地说只是一个联系。与上述相关的问题是试图理解流形上的过程结构，并了解如何模拟它们。我们将首先关注最简单的曲面流形——任意维的单位球——以及它的正则过程——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。我们特别感兴趣的是所谓的斜积分解它将一个过程分解成两个更小维度且表现良好的过程，通常发生在给定的度量不是乘积的积空间中，而是所谓的扭曲积空间。使用这种分解，理论上可以将过程简化为一系列相关的一维过程，这些过程很容易理解。特别是，通常可以模拟一维过程，而将分解反过来可以为原始多维过程产生有用的模拟算法。我们希望解决的另一个问题是如何以规范和合适的方式定义一般流形上的某类过程-列维过程。列维过程最初是在欧几里得空间上定义的，一个特别有趣的特征是它们本质上是最简单的一类表现跳跃的过程。虽然大多数理论转化为李群设置（欧几里得空间尤其也是李群），但对于一般流形可以做什么知之甚少。其中一个问题是，增量的概念在一般流形上没有意义，而跳跃的添加是另一个问题，因为理论上跳跃可以带我们在流形上的任何地方，流形通常只在局部表现良好，而全局结构可能非常复杂。在流形上定义一些列维过程方面已经做了一些工作，但似乎还有改进要做，因为似乎存在更多的过程，这些过程将被正确地称为列维过程，并且没有包括在以前的结构中。我们的目标是使用微分几何中的某些工具-主纤维束，连接，框架束，（反）发展-解决这个问题，并尝试对过程的最大类别进行分类，这些过程可以被正确地称为流形上的列维过程，当我们考虑欧几里德空间和李群时，这个概念应该产生列维过程的经典概念，此外，我们应该在这个设置中得到所有可能的列维过程（这不是当前结构的情况）。

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).