Stochastic Analysis: Theory and Applications

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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.