Stochastic Analysis: Theory and Applications

PI将研究在应用科学和数学研究启发的数学模型研究中出现的基础问题。具有惰性漂移的反射布朗运动可以作为阿基米德原理等物理现象的数学模型。带斜反射的反射布朗运动可以用来模拟繁忙交通下的排队网络，也可以用来求解混合边界条件下的偏微分方程。在马氏过程边界理论的研究中，自然会出现带织补的布朗运动。这类扩散过程是研究多连通平面区域上共形映射和Komatu-Loewner方程的有效工具。PI的将开发这样的系统和过程的数学理论。探讨了变形介质中随机游动不变性原理和区域随机均匀化的新途径。本课程将探讨光在粗糙表面上的反射模型及比拉德散射。 PI将研究害羞耦合和一些Fleming-Viot型模型。本文将系统地研究类稳定过程的边界Harnack原理和旋转对称Levy过程的双边Dirichlet热核估计，并以各种方式介绍他们在随机分析方面的进展。现有的非常活跃的数学系概率研讨会在华盛顿大学吸引了高度多样化的观众-研究人员在数学，统计，应用数学，代表从一个研究小组在一家私营公司，博士后学员和研究生。妇女作为发言者和与会者参加研讨会的人数很多。每年一次，为期一天的西北概率研讨会，汇集了来自俄勒冈州，不列颠哥伦比亚省和华盛顿感兴趣的人。同样，女性对这项年度活动的参与意义重大。PI积极参与随机分析及其应用国际会议的组织工作，这是自2006年以来定期（几乎每年）举办的年度活动。它汇集了来自世界各地的随机分析和相关领域的专家，以调查该领域，交流思想并促进未来的合作。青年研究人员和博士生的参与意义重大。华盛顿大学设立了一个计算金融学项目。PI计划参与该计划，以帮助向非数学受众传播随机分析的知识，包括该领域的最新进展，并强调学生。在过去，该计划吸引了来自不同部门的学生，如数学，统计，经济，金融，电气工程，物理和计算机科学。PI目前有四名博士生，包括一名女性和两名非白人学生。所有博士生目前都正在或预计将积极参与本提案中概述的研究。

The PI's will study foundational questions that arise in the study of mathematical models inspired by applied science and mathematical research. Reflected Brownian motion with inert drift can be used as a mathematical model for some physical phenomena such asArchimedes' principle. Reflected Brownian motion with oblique reflection can be used to model queuing networks under heavy traffic and to solve partial differential equations with mixed boundary conditions. Brownian motion with darning arises naturally in the study of boundary theory of Markov processes. This class of diffusion processes are an effective tool in the study of conformal mappings and Komatu-Loewner equations in multiply connected planar domains. The PI's will develop the mathematical theory of such systems and processes. New approach to invariance principle of random walks in distorted medium and stochastic homogenization in domains will be explored. Models of light reflected on rough surfaces and billard scattering will be investigated. The PI's will study shy couplings and some Fleming-Viot-type models. Boundary Harnack principle for stable-like processes and sharp two-sided Dirichlet heat kernel estimates for rotationally symmetric Levy processes will be systematically investigated.The PI's will disseminate the information on their advances in stochastic analysis in various ways. The existing very active Department of Mathematics probability seminar at the University of Washington attracts highly diverse audience - researchers in mathematics, statistics, applied mathematics, representatives from a research group at a private corporation, post-doctoral trainees and graduate students. There is a significant participation of women in the seminar, both as speakers and attendees. Once a year, the Northwest Probability Seminar, a one-day conference, brings together interested people from Oregon, British Columbia and Washington. Again, the participation of women in this annual event is significant. The PI's are strongly involved in the organization of the International Conference on Stochastic Analysis and its Applications, an annual event that has been held regularly (almost every year) since 2006. It brings experts in stochastic analysis and related fields from all over the world to survey the field, exchange ideas and to foster future collaborations. Participation of young researchers and Ph.D students is significant. The University of Washington has established a Program in Computational Finance. Both PI's plan to participate in the program to help disseminate the knowledge of the stochastic analysis, including the most recent advances in the field, to non-mathematical audiences, with stress on students. In the past, the program attracted students from diverse departments, such as Mathematics, Statistics, Economics, Finance, Electrical Engineering, Physics and Computer Science. The PI's have four doctoral students at the moment, including one woman and two non-white students. All doctoral students currently are or are expected to be actively involved in the research outlined in this proposal.