Foundations and Applications of Stochastic Analysis

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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. One of the problems, the so-called third boundary problem, is a mathematical model for real systems with semipermeable membranes. The PI's will develop the mathematical theory of such systems with fractal structure. The quality of approximation of continuous processes by discrete models will be investigated. Simple stochastic processes, such as Brownian motion, can be perturbed in various ways, for example, by inert drift or oblique reflection. These more complicated models require new mathematical tools which will be developed. The PI's will study some multiparticle systems confined to bounded domains. Several mathematical tools will be sharpened, including new versions of the boundary Harnack principle and heat kernel estimates.The project will develop theoretical mathematical tools needed in and inspired by scientific application of mathematics.The applied phenomena that gave rise to the ideas under consideration include natural semipermeable membranes, such as human lungs, and man-made objects, such as catalytic converters and car batteries.Some of the applications are related to computer science, especially approximation of real continuous phenomena with digital models suitable for computer based analysis. The PI's will also study mathematical models describing populations undergoing evolution according to various schemes.