Stochastic Processes

Marcus教授Marcus和Rosen将继续研究对称马尔可夫过程泛函的连续性及其极限行为。利用Dykin的同构定理和他们所得到的推广，他们将利用高阶Wick幂和Wick积高斯混沌过程与一大类Levy过程的重整化自交局部时之间的相互作用，研究直线和曲线上的Levy过程的“近交”现象和Levy过程的局部时的路径性质.所考虑的过程包括一维、二维和三维的布朗运动和稳定过程，以及它们的吸引域中的过程。Marcus和Rosen计划将自交局部时间泛函刻画为相关Dirichlet过程分解的零二次变差过程。假设自交局部时间泛函的二次变化为零，自然会问它的精确p-变化是多少。他们也计划调查这个问题。他们希望能够使用基于随机傅里叶级数理论的早期论文中开发的技术，通过用分布代替测量来扩展自相交局部时间泛函的定义。他们将继续研究高阶高斯混沌过程和平稳移动平均的连续性，以及绝对连续的无限可分过程的水平交叉。他们还将研究四维格子上随机游动的占用测度和交集的多重分形谱。这项研究涉及随机过程的基本性质。它揭示了马尔可夫过程和高斯混沌过程这两类以前被认为是不相关的随机过程之间的深层联系。这种关系增加了对这两个组成部分过程的理解，这种理解不能通过单独考虑这两个过程来获得。这些结果可能在数据的发送、接收和分析、量子物理和金融衍生品分析中有广泛的应用。

9802753MarcusProfessors Marcus and Rosen will continue their study of continuity properties of functionals of symmetric Markov processes and their limiting behavior. Using an isomorphism theorem of Dynkin and extensions that they have obtained they will exploit the interplay between high order Wick power and Wick product Gaussian chaos processes and the renormalized self-intersection local time of a large class of Levy processes, to study the phenomenon of ``near-intersections'' of a Levy process and path properties of local times of Levy processes on lines and curves. The processes considered include Brownian motion and stable processes in dimensions one two and three, as well as processes in their domains of attraction. Marcus and Rosen plan to characterize the self-intersection local time functional as the process of zero quadratic variation in the decomposition of an associated Dirichlet process. Given that the self-intersection local time functional has zero quadratic variation it is natural to ask what is its precise p-variation. They plan to investigate this question as well. They expect to be able to use techniques developed in an earlier paper based on the theory of random Fourier series to expand the definition of the self-intersection local time functional by replacing measures by distributions. They will continue their study of continuity properties of high order Gaussian chaos processes and stable moving averages, and level crossings of absolutely continuous infinitely divisible processes. They will also study the multifractal spectrum of occupation measures and intersections of random walks on a four dimensional lattice.This research deals with fundamental properties of stochastic processes. It exposes a deep relationship between two classes of stochastic processes, Markov processes and Gaussian chaos processes that previously were considered to be unrelated. This relationship leads to increased understanding of the two component processes, an understanding that cannot be obtained by considering the processes separately. These results may have long range applications in the transmission, reception and analysis of data, in quantum physics and in the analysis of financial derivatives.