Stochastic Evolution Equations Driven by Nonmartingale Random Fields and Related Topics

随着越来越多地使用大型遥感图像和视频流储存库，对时空现象的分析变得越来越重要。事实上，现代社会对复杂空间系统时间演化分析的需求远远超过了迄今为止所取得的技术、工程和数学进步。在实际应用中遇到的空间结构往往是不平凡的，由于长的内存，或由于存在的“粗糙”的几何约束，现有的分析工具无法描述时空演变的系统，必要的参数估计和预测问题的解决方案非常具有挑战性。这个项目解决了其中的一些挑战，并侧重于分析连续随机演化方程驱动的沃尔泰拉随机场，其中缺乏鞅和马尔可夫结构。具体而言，研究将涉及以下主要领域：1。多参数沃尔泰拉随机场的分析（这包括强/弱鞅变换生成相同的自然过滤的沃尔泰拉领域的建设，研究集成的沃尔泰拉内核相对于非高斯鞅，分析沃尔泰拉随机场与无限维参数空间，有效的模拟技术的发展沃尔泰拉多参数领域）; 2.沃尔泰拉随机场随机微积分的发展和沃尔泰拉过程局部时的研究; 3.分析由沃尔泰拉随机场驱动的随机演化方程（包括研究受沃尔泰拉型噪声扰动的抛物型随机偏微分方程，分析当观测噪声具有沃尔泰拉结构时随机场非线性滤波中产生的随机演化方程，在这种情况下次优滤波器的表示和构造，开发对具有长记忆空间结构的图像进行“去噪”的数值技术。随机演化方程所产生的相互作用的随机运动的粒子，这是受Volterra型噪声系统，也将进行研究。4.推导由沃尔泰拉场驱动的演化方程（当沃尔泰拉核已知时的系数估计，加上沃尔泰拉核本身的参数估计）。预计这些结果将在各种复杂的空间系统中有用，因为它们会随着时间的推移而演变。该项目有一个实质性的非线性滤波组件，它自然享有在大量的设置中的图像/视频“去噪”是重要的应用程序。从通过卫星跟踪飓风到涉及分析由各种设备记录和传输的空间数据流的医疗程序，我们每天都被复杂的系统所包围，并且充斥着空间数据。随着在地质和天体物理学、气候学、生物医学应用和人口动态研究等不同领域越来越多地使用遥感图像和视频流，空间现象分析和建模的重要性日益增加。具体应用包括移动商务行业（基于位置的服务）、美国航天局关于厄尔尼诺气候影响的研究、利用卫星图像进行的土地使用分类和全球变暖、通过远程和地面望远镜分析恒星和星系的演变、国家卫生研究所关于预测疾病传播和流行病控制的研究，更不用说根据专门地图对交通和基础设施趋势进行的“例行”分析，这些地图代表在特定时间点拍摄的复杂空间动态系统状态的嘈杂"快照”。使用随机场，它允许考虑复杂系统中变量之间的空间相互作用，是一个越来越重要的工具，用于统计力学，空间统计，神经网络建模等许多问题。与此同时，上述许多现实生活中的应用缺乏数学上方便的马尔可夫和鞅（空间）结构，使大多数可用的分析工具不足。该项目旨在解决这些挑战，并开发一个理论的随机演化方程驱动的一大类随机场，称为沃尔泰拉随机场，其中缺乏鞅和马尔可夫结构，并允许广泛的（长和短）的内存和路径属性。该项目还有一个重要的非线性滤波组件，它自然在空间滤波和图像和视频去噪非常重要的大量科学和商业环境中得到应用。这项资助的研究成果旨在推进随机分析的当前最新水平，并将促进其他领域科学家使用概率方法。这鼓励了不同背景的研究人员之间的互动，并最终导致在每个领域的新想法，技术和结论。例如，复杂介质，其重要的应用和潜在的微观过程，通常与长期记忆，长程相互作用和非马尔可夫动力学。后者的一些例子包括许多耦合元素系统中的过程、胶体聚集体和化学反应介质、多孔介质、量子力学和量子场论、等离子体物理学、磁层和许多其他领域。该项目还包含大量教育内容。

Analysis of spatio-temporal phenomena is becoming important with the increasing use of large repositories of remote-sensing images and video-streams. In fact, modern society's need for analysis of temporal evolution of complex spatial systems far outpaces the technological, engineering and mathematical advances made so far. Spatial structures encountered in actual practice are often nontrivial due to long memory, or due to presence of "rough" geometrical constraints, and the existing analytical tools fail to describe spatio-temporal evolution of such systems, making solution of necessary parameter estimation and prediction problems very challenging. This project addresses some of these challenges and focuses on the analysis of continuous stochastic evolution equations driven by Volterra random fields, which lack martingale and Markov structures. More specifically, the research will deal with the following major areas: 1. Analysis of multiparameter Volterra random fields (this includes construction of strong/weak martingale transforms generating the same natural filtration as the Volterra field, study of integrated Volterra kernels with respect to non-Gaussian martingales, analysis of Volterra random fields with infinite-dimensional parameter spaces, development of efficient simulation techniques for Volterra multiparameter fields); 2. Development of stochastic calculus with respect to Volterra random fields and the study of local time for Volterra processes; 3. Analysis of stochastic evolution equations driven by Volterra random fields (including the study of parabolic SPDEs perturbed by Volterra-type noise, analysis of stochastic evolution equations arising in nonlinear filtering of random fields when the observation noise has Volterra structure, representations and construction of suboptimal filters in that setting, development of numerical techniques for ``denoising" of images with long-memory spatial structure. Stochastic evolution equations arising from systems of interacting randomly moving particles, which are subject to Volterra-type noises, will also be studied. 4. Inference for evolution equations driven by Volterra fields (estimation of coefficients when the Volterra kernel is known, plus estimation of parameters of the Volterra kernel itself). The results are expected to be useful in a wide variety of complex spatial systems, as they evolve in time. The project has a substantial nonlinear filtering component, which naturally enjoys applications in a great number of settings where image/video ``denoising" is important. These range from tracking hurricanes via satellites to medical procedures involving analysis of spatial data streams that are recorded and transmitted by various devices.Everyday we are surrounded by complex systems and are awash in spatial data. The importance of analysis and modelling of spatial phenomena is growing with the increasing use of remote-sensing images and video-streams arising in such diverse areas as geological and astrophysical sciences, climatology, biomedical applications and studies of population dynamics. Specific applications include Mobile-commerce industry (location based services), NASA's study of climatological effects of El Nino, land-use classification and global warming using satellite imagery, analysis of evolution of stars and galaxies via remote and Earth-based telescopes, studies by National Institute of Health on predicting spread of disease and epidemic control, not to mention ``routine" analysis of traffic and infrastructure trends on the basis of specialized maps, which represent noisy ``snapshots" of the state of complex spatial dynamical system taken at particular points in time. The use of random fields, which allow to take into account spatial interactions among variables in complex systems, is an increasingly important tool used in numerous problems of statistical mechanics, spatial statistics, neural network modelling, and others. At the same time many of the above real-life applications lack mathematically convenient Markov and martingale (spatial) structures, making the majority of available analytical tools inadequate. This project aims to address these challenges and to develop a theory of stochastic evolution equations driven by a large class of random fields, called Volterra random fields, which lack martingale and Markov structures and allow for a wide range of (both, long and short) memory and pathwise properties. The project also has a significant nonlinear filtering component, which naturally enjoys applications in a great number of scientific and commercial settings where spatial filtering and image and video denoising are important. The research results emanating from this grant aim to advance the current state-of-the-art in stochastic analysis and will promote use of probabilistic methods among scientists in other areas. This encourages interaction between researchers with varying backgrounds and ultimately leads to new ideas, techniques and conclusions in each of those fields. For example, complex media, with its important applications and underlying microscopic processes, is typically linked to long-term memory, long-range interactions and non-Markovian kinetics. Some of the examples of the latter include processes in systems of many coupled elements, colloidal aggregates and chemical reaction medium, porous media, quantum mechanics and quantum field theory, plasma physics, magnetosphere and many other fields. The project also has a substantial educational component.