Stochastic processes on curved spaces

弯曲空间上的随机过程

• 批准号: 1948092
• 金额: --
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1948092
• 关键词: Stochastic; processes; curved; spaces

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).

摘要：随机过程在（平坦的）欧几里得空间上得到了很好的理解。但最近有一些有趣的过程，其状态空间是在任意（弯曲）流形上。半鞅的概念在流形上有很好的定义，只要流形给出了黎曼度量，或者更一般地说只是一个连接，流形上的半鞅和欧氏空间上的半鞅之间实际上有一个简洁的连接。与上述相关的问题是试图理解流形上过程的结构，看看它们如何被模拟。我们将首先关注最简单的弯曲流形-任意维的单位球面-和它的正则过程-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个过程起着关键作用。我们希望利用这一事实来获得这一过程的结构性结果。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少维度和行为良好的过程，通常发生在乘积空间上，其中给出的度量不是乘积，而是所谓的扭曲乘积。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，这些过程是很好理解的。特别是人们通常可以模拟一维过程和扭转的分解周围可能会产生有用的模拟算法为原来的多维process.Another问题，我们希望解决的是如何定义在一个规范的和适当的方式在一般流形上的某一类过程- 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最简单的一类表现跳跃的过程。虽然大多数理论都转化为李群结构（特别是欧几里得空间也是李群），但对一般流形上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，另一个问题是跳跃的增加，因为跳跃理论上可以带我们到流形上的任何地方，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Levy过程已经做了一些工作，但似乎还需要改进，因为似乎存在更多的过程，这些过程将被正确地称为Levy过程，并且不包括在以前的构造中。我们的目标是使用微分几何中的某些工具--主纤维丛、联络、框架丛，（反）发展-为了解决这个问题，并试图分类最大类的过程，这类过程可以被正确地称为流形上的Levy过程，当我们考虑欧氏空间和李群时，这个概念应该产生Levy过程的经典概念，此外，我们应该得到所有可能的Levy过程，这种设置（这不是当前构造的情况）。