Topics in Stochastic Analysis

University Of Washington, Seattle WA

Investigators

Abstract

The PI's will investigate several problems in stochastic analysis of pure and applied nature. Competing particle systems appear in various scientific models including biology and theory of combustion. The PI's will study various aspects of such models, including hydrodynamic limits, tagged particle behavior and properties of the associated partial differential equations. Long term behavior of reflected Brownian motions driven by the same noise will be examined. Geometric properties of Neumann eigenfunctions will be investigated. Uniqueness of solutions to some stochastic differential equations will be proved. Boundary behavior of harmonic functions and local and global behavior of heat kernel associated with jump processes will be examined. Brownian motion on a fractal set will be investigated. Inverse problem is a mathematical model arising naturally in applied sciences from geophysics to medicine and physics. The PI's will study the foundational questions related to this model, such as existence and uniqueness of the solutions to the corresponding equations. Other models and processes will be the subject of separate but related studies; they include Markov processes conditioned on a small time scale, a change of variable formula for processes with 4-th order scaling properties and non-local operators of variable order. Brownian motion and Markov processes are models for a wide range of natural phenomena, such as weather and climate, various functions of living organisms and financial markets. One of the scientific goals is to make accurate predictions based on available data. Mathematical methods have to be developed to make such predictions possible and reliable. The PI's will work on both theoretical questions leading to general results and on specific models that can serve as testbeds for the general methods. The results of theoretical research are often used to make quantitative predictions in well undrstood models and to make qualitative predictions related to complex systems. The research of PI's will be mostly focused on processes that display no or little long time memory. Such systems are popular models for inanimate matter and man-made systems, but also for many biological systems.