The PIs investigate several problems in stochastic analysis of both a pure and applied nature. Inverse problems and the third boundary problem are mathematical models arising naturally in applied sciences from geophysics to medicine and physics. The PIs study the foundational questions related to these models, such as existence and uniqueness of the solutions to the corresponding equations, and also address specific problems in each of these fields. The PI's investigate several aspects of reflected Brownian motion, including the coalescence of synchronous couplings, reflection with inert drift, existence of shy couplings, construction of reflected processes via myopic conditioning, and the pathwise uniqueness for the Skorohod equation. Other models and processes are the subject of separate but related studies; they include the skew Brownian motion stochastic flow, censored stable processes, and jump-type processes. The PI's attack some well known "open problems" on Neumann eigenfunctions and reflected Brownian motion. The project is devoted to modeling, at the mathematical level, of natural, technical, economic and social phenomena involving randomness. The field of stochastic analysis arose as a need to understand phenomena as diverse as the stock market, non-destructive analysis of materials, medical imaging, and heat dissipation. The mathematical side of the project provides a foundation for the applications such as computer programs that help in real life situations. The applied sciences routinely use mathematical formulas of stochastic analysis. The project aims at providing usable information about stochastic processes.
