Stochastic Analysis: Theory and Applications

PI将研究受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。带有惰性漂移的反射布朗运动可以作为一些物理现象的数学模型，如阿基米德原理。具有斜反射的反射布朗运动可以用于大流量下排队网络的建模和求解具有混合边界条件的偏微分方程。在研究马尔可夫过程的边界理论时，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域上共形映射和Komatu-Loewner方程的有效工具。PI将发展这些系统和过程的数学理论。探讨了畸变介质中随机漫步的不变性原理和域内随机均匀化的新方法。光在粗糙表面上的反射模型和广告牌散射将被研究。PI将研究害羞耦合和一些弗莱明-维特模型。系统地研究了类稳定过程的边界哈纳克原理和旋转对称Levy过程的尖锐双面Dirichlet热核估计。PI将以各种方式传播他们在随机分析方面取得的进展。华盛顿大学现有的非常活跃的数学系概率研讨会吸引了各种各样的听众——数学、统计学、应用数学方面的研究人员、私营公司研究小组的代表、博士后学员和研究生。妇女作为发言者和与会者大量参加了讨论会。西北概率研讨会（Northwest Probability Seminar）每年举行一次，为期一天，汇集了来自俄勒冈州、不列颠哥伦比亚省和华盛顿州的相关人士。同样，妇女参与这一年度活动意义重大。PI积极参与组织国际随机分析及其应用会议，这是自2006年以来定期（几乎每年）举行的年度活动。它汇集了来自世界各地的随机分析和相关领域的专家来调查该领域，交流思想并促进未来的合作。年轻研究人员和博士生的参与是重要的。华盛顿大学建立了一个计算金融项目。两个PI都计划参与这个项目，以帮助传播随机分析的知识，包括该领域的最新进展，给非数学的观众，重点是学生。过去，该项目吸引了来自不同院系的学生，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.