Constrained random phenomena using rough paths

随机过程在自然界和人造世界中普遍存在。这些过程通常被观察、测量或体验为在某个状态空间中随时间演变的路径。噪声的影响常常使这些进化路径的轨迹变得不可预测和高度不稳定。在简单的例子中，例如股票价格的变动，状态空间可能是非负实数。在更一般的例子中，演化的动力学可能是受约束的，例如刚体的运动，或者地球大气层的一部分中的粒子的演化。该项目将为分析弯曲空间上的高能随机轨迹开发广泛的建模框架。这其中的一个关键因素将是使用莱昂斯的粗糙路径分析。上个世纪对布朗运动的精确研究在自然现象的模拟方面取得了壮观的结果。黎曼布朗运动的Eells-Elworth-Malliavin整体构造极大地推动了我们对流形布朗运动的理解。然而，人们现在越来越清楚地认识到，基于布朗运动的模型并不总是合适的；持久性、长期依赖性和动量是互联网流量排队网络、水文学和市场价格波动中长期观察到的行为特征。布朗运动属于统计学中一类基本的随机过程，称为高斯过程。这一类既简单易用，又足够广泛，足以捕捉演化系统中的随机记忆效应。在这个项目中，将结合随机分析、概率、粗糙路径理论和随机微分几何的技术，以精确和定量的方式研究黎曼流形上一类高斯过程的性质。我们期望在这个过程所处的空间的随机性和几何形状之间会有有趣的相互作用。该项目的一个关键目标将是向更广泛的科学界提供更深入的理解和技术，供他们在工作中使用。

Random process are ubiquitous throughout the natural and man-made world. These processes are often observed, measured or experienced as paths evolving over time in some state space. The effect of noise often makes the trajectory of these evolving paths unpredictable and highly erratic. In simple examples, such as the movement of a share price, the state space might be the non-negative real numbers. In more general examples the dynamics of the evolution may be constrained, such as the movement of a rigid body, or the evolution of particles in a section of the earth's atmosphere. This project will develop broad modelling framework for the analysis of highly-energetic random trajectories on curved spaces. A key ingredient of this will be the use Lyons' rough path analysis. The precise study of Brownian motion over the last century has led to spectacular results in the modelling of natural phenomena. Our understanding of manifold-valued Brownian motion was given great impetus by the Eells-Elworthy-Malliavin global construction of Riemannian Brownian motion. It is now however increasingly well understood that model based on Brownian motion are not always appropriate; persistence, long-time dependence and momentum are long-observed features of behaviour in queueing networks for internet-traffic, in hydrology, and in the fluctuation of market prices. Brownian motion belongs to a fundamental class of random processes in statistics called Gaussian processes. This class is both simple enough to work with, and broad enough to capture random memory-effects in evolving systems.In this project will will combine techniques from stochastic analysis, probability the theory of rough paths and stochastic differential geometry to study, in a precise and quantitative way, properties of a class of Gaussian processes on Riemannian manifolds. We expect there to be interesting interplay between the randomness and the geometry of the space which the process inhabits. A key objective of the project will be to furnish the wider scientific community with deeper understanding and techniques which they can utilise in their work.