Stochastic Evolution Equations Driven by Nonmartingale Random Fields and Related Topics

随着大量遥感图像和视频流的使用日益增多，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的时间演化分析的需求远远超过了迄今为止在技术、工程和数学方面取得的进步。在实际应用中遇到的空间结构往往是非平凡的，这是由于长时间的记忆，或由于存在“粗糙”的几何约束，而现有的分析工具无法描述这类系统的时空演化，这使得解决必要的参数估计和预测问题非常具有挑战性。这个项目解决了其中的一些挑战，并专注于分析由Volterra随机场驱动的连续随机发展方程，这些随机场缺乏鞅和马尔可夫结构。更具体地说，这项研究将涉及以下主要领域：1.多参数Volterra随机场的分析(这包括构造产生与Volterra场相同的自然滤波的强/弱鞅变换、关于非高斯鞅的积分Volterra核的研究、具有无限维参数空间的Volterra随机场的分析、Volterra多参数场的有效模拟技术的发展)；2.关于Volterra随机场的随机演算的发展和Volterra过程的局部时间的研究；3.Volterra随机场驱动的随机演化方程的分析(包括受Volterra型噪声扰动的抛物型SPDEs的研究，当观测噪声具有Volterra结构时，随机场的非线性滤波所产生的随机演化方程的分析，该背景下次优滤波器的表示和构造，具有长记忆空间结构的图像的去噪的数值技术的发展。还将研究由受Volterra型噪声影响的随机运动粒子系统相互作用而产生的随机演化方程。4.由Volterra场驱动的演化方程的推断(已知Volterra核时的系数估计，加上Volterra核本身的参数估计)。随着时间的推移，这些结果有望在各种复杂的空间系统中发挥作用。该项目有一个重要的非线性过滤组件，它自然适用于图像/视频‘去噪’非常重要的大量环境中。从通过卫星跟踪飓风到医疗程序，这些程序涉及分析由各种设备记录和传输的空间数据流。每天，我们都被复杂的系统包围，空间数据泛滥。随着越来越多地使用在地质和天体物理科学、气候学、生物医学应用和人口动态研究等不同领域产生的遥感图像和视频流，对空间现象进行分析和建模的重要性日益增加。具体应用包括移动电子商务行业(基于位置的服务)、美国航天局利用卫星图像对厄尔尼诺现象的气候学影响、土地使用分类和全球变暖进行的研究、通过远程望远镜和地面望远镜分析恒星和星系的演变、国家卫生研究所进行的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“常规”分析，这些地图代表了在特定时间点拍摄的复杂空间动力系统状态的嘈杂“快照”。随机场的使用允许考虑复杂系统中变量之间的空间相互作用，这是在统计力学、空间统计、神经网络建模等许多问题中使用的日益重要的工具。同时，上述现实生活中的许多应用缺乏数学上方便的马尔可夫和鞅(空间)结构，这使得大多数可用的分析工具都不够用。这个项目的目的是解决这些挑战，并发展一种由一大类随机场驱动的随机演化方程理论，称为Volterra随机场，它缺乏鞅和马尔可夫结构，并允许广泛的(长和短)记忆和路径性质。该项目还具有重要的非线性滤波组件，它自然适用于大量科学和商业环境，其中空间滤波和图像和视频去噪非常重要。这笔赠款产生的研究成果旨在促进当前随机分析的最新水平，并将促进其他领域的科学家使用概率方法。这鼓励了具有不同背景的研究人员之间的互动，并最终在每个领域产生了新的想法、技术和结论。例如，复杂的介质及其重要的应用和潜在的微观过程，通常与长期记忆、长期相互作用和非马尔科夫动力学有关。后者的一些例子包括许多耦合元素、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、磁层和许多其他领域中的过程。该项目还包含大量的教育内容。

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.