Mathematical Sciences: Stochastic Processes

摘要研究者将继续研究马尔可夫过程的加性泛函的连续性及其极限行为。利用Dynkin的同构定理以及巴拿赫空间和随机傅立叶级数的概率论的结果，他们将探索马尔可夫过程和高斯过程的连续加性泛函以及混沌和随机傅立叶级数之间的关系。通过这种方式，他们希望能够使用关于每个不同过程的已知结果来获得关于其他过程的新结果，就像他们在局部时间和高斯过程的研究中所做的那样。他们还计划使用他们早期工作中开发的技术来研究随机漫步的交叉点数量，特别是在临界和超临界情况下。考虑到与tm Wick平方相关的一类二阶高斯混沌在正连续加性泛函研究中的重要性，他们将试图找到这些混沌的连续性的充分必要条件。他们还将尝试使用超过程的自碰撞局部时间来构建自相互作用的测度值扩散，并研究超布朗运动的加性泛函的大偏差。这项研究涉及随机过程的基本性质，在处理随机现象的所有领域都有潜在的应用。调查人员将考虑以随机方式随时间演变的数据，比如从遥远的卫星接收到的电信号或金融交易中的美元价值。特别是，他们将考虑数据取特定值或在一个小的、重加权的值范围内的时间量，以及该时间量如何随着特定值集的变化而变化。他们计划将这种方法应用于一系列被称为Levy过程的过程中。这些过程通常被认为是许多重要实际应用的模型，但到目前为止，它们被认为很难进行深入分析。作者计划使用他们自己和其他人最近开发的几种新的和强大的方法来完成这种深入的分析。

9503519 Marcus Abstract The investigators will continue their study of continuity properties of additive functionals of Markov processes and their limiting behavior. Using an isomorphism theorem of Dynkin and results from the theory of probability in Banach spaces and random Fourier series they will explore relationships between continuous additive functionals of Markov processes and Gaussian processes and chaoses and random Fourier series. In this way they expect to be able to use known results about each of the different processes to obtain new results about the other ones as they did in their work on local times and Gaussian processes. They also plan to use techniques developed in their earlier work to study the number of intersections of random walks, particularly in the critical and super-critical cases. Motivated by the importance of a special class of second order Gaussian chaoses related tm Wick squares in the study of positive continuous additive functionals, they will try to find necessary and sufficient conditions for the continuity of these chaoses. They will also attempt to use self-collision local times of superprocesses to construct a self-interacting measure valued diffusion and study the large deviations of additive functionals of super Brownian motion. This research deals with fundamental properties of stochastic processes and has potential application in all areas that deal with random phenomena. The investigators will consider data that evolves in time in a random fashion, such as electrical signals received from distant satellites or dollar values in financial transactions. In particular they will consider the amount of time the data takes specific values or is in a small, heavily weighted, range of values and how this amount of time varies as the set of specific values vary. They plan to do this for a wide class of processes called Levy processes. These processes have often been thought of as models for many important practical applications but, so far, they have b een considered too difficult to analyze very deeply. The authors plan to use several new and powerful methods, recently developed by themselves and others, to accomplish this deep analysis.