Topics in Stochastic Analysis

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The PI's will investigate several problems in stochastic analysisof pure and applied nature. Competing particle systems appear invarious scientific models including biology and theory ofcombustion. The PI's will study various aspects of such models,including hydrodynamic limits, tagged particle behavior andproperties of the associated partial differential equations. Longterm behavior of reflected Brownian motions driven by the samenoise will be examined. Geometric properties of Neumanneigenfunctions will be investigated. Uniqueness of solutions tosome stochastic differential equations will be proved. Boundarybehavior of harmonic functions and local and global behavior ofheat kernel associated with jump processes will be examined.Brownian motion on a fractal set will be investigated. Inverseproblem is a mathematical model arising naturally in appliedsciences from geophysics to medicine and physics. The PI's willstudy the foundational questions related to this model, such asexistence and uniqueness of the solutions to the correspondingequations. Other models and processes will be the subject ofseparate but related studies; they include Markov processesconditioned on a small time scale, a change of variable formulafor processes with 4-th order scaling properties and non-localoperators of variable order.Brownian motion and Markov processes are models for a wide rangeof natural phenomena, such as weather and climate, variousfunctions of living organisms and financial markets. One of thescientific goals is to make accurate predictions based onavailable data. Mathematical methods have to be developed to makesuch predictions possible and reliable. The PI's will work on boththeoretical questions leading to general results and on specificmodels that can serve as testbeds for the general methods. Theresults of theoretical research are often used to makequantitative predictions in well undrstood models and to makequalitative predictions related to complex systems. The researchof PI's will be mostly focused on processes that display no orlittle long time memory. Such systems are popular models forinanimate matter and man-made systems, but also for manybiological systems.