Infinite dimensional analysis, viscosity solutions and applications

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The research of the project is focused on fully nonlinear first- and second-order partial differential equations (PDE) in infinite dimensional spaces and applications thereof. Primary examples of such PDE are equations of Hamilton-Jacobi-Bellman-type (HJB) that are associated with optimal control of deterministic and stochastic PDE. The theory of PDE in infinite dimensional spaces has been studied from the point of view of mild, regular, weak, and viscosity solutions and has established itself over the last two decades as one of the modern tools of infinite dimensional analysis. In particular, in addition to optimal control of PDE, the theory of viscosity solutions of such equations has found applications in areas such as the study of large deviations of infinite dimensional diffusions, the theory of bond markets and other aspects of mathematical finance, and stochastic invariance. In this project the principal investigator focuses on two emerging areas of infinite dimensional PDE that are wide open. The first is integro-PDE in Hilbert spaces. The goal is to develop a viscosity solution theory for fully nonlinear first- and second-order integro-PDE in Hilbert spaces and study its various applications. The interest in such equations comes primarily from their association with infinite dimensional jump-diffusion processes, in particular with stochastic PDE driven by Levy processes. Regarding applications, the theory will be used to develop a PDE approach to large deviations for solutions of stochastic PDE with small Levy noise intensity and to study infinite dimensional Black-Scholes and Black-Scholes-Barenblatt integro-PDE coming from the theory of bond markets driven by impulsive noise. A second area of emphasis is PDE in the space of probability measures, which finally seems open for development owing to recent advances in the theory of mass transport and abstract gradient and Hamiltonian flows. This new area is extremely interesting and important, and the project will concentrate on its applications to large deviations and statistical mechanics. Other problems contained in the project include the use of HJB equations to obtain necessary and sufficient conditions for optimality for optimal control of PDE and investigations into maximum principles.The project contains a pioneering program of research that is aimed at the development of new tools in partial differential equations and infinite dimensional analysis. The proposed research spans areas as diverse as nonlinear partial differential equations, functional analysis, probability, stochastic processes, stochastic partial differential equations, mathematical finance, optimal control, game theory, statistical mechanics, mass transport, and calculus of variations. In particular, it will provide analytical techniques for the study of infinite dimensional jump diffusion processes that are used in stochastic modeling of various phenomena in which random and violent events can occur. Beyond mathematics, the project should have impact and stimulate research in fields such as engineering, physics, finance, and economics. The broader impacts of the project will also include attracting and training graduate students and postdoctoral scholars.