This research project is concerned with partial differential equations of elliptic and parabolic type, especially equations with random coefficients. There are deep connections between equations with random coefficients and statistical mechanics, in particular particle systems. In recent work the principal investigator and co-authors have investigated connections between the one dimensional nonlinear stochastic Fisher-Kolmogorov-Petrowski-Piscunov (FKPP) equation and certain particle systems. Their work introduced new techniques which, if more fully developed, should yield the proof of some conjectures of physicists concerning the wave speed of the stochastic FKPP equation. A major goal of the project is to obtain the proof of some of these conjectures. In another direction the PI plans to continue his research on linear stochastic elliptic and parabolic equations, in particular with the equations governing random walk in random environment. He has already established some unexpected connections between this problem and certain results in combinatorics related to graph connectivity. These arose in the study of the formal perturbation theory for the problem. The main goal in the current project is to prove an inequality for the effective diffusivity constant. He expects to uncover some new connections with other areas in the study of this problem also. The final part of the project is concerned with problems in partial differential equations which occur in finance. The principal investigator plans to continue working with his graduate student on a model for efficient management of an insurance company. Mathematically the problem is a problem of stochastic control theory. A satisfactory analysis of the problem will include existence and uniqueness proofs and a detailed study of the free boundary which occurs.
The purpose of this research project is the rigorous mathematical analysis of certain partial differential equations. These equations govern the behavior of many processes which occur in physics and engineering. During the last thirty years they have also been used in finance in the systematic study and pricing of financial instruments which depend on market volatility, such as options. The principal investigator plans to do an in depth study of some problems related to physics-engineering applications and also some problems with applications in finance. The financial application is concerned with a simple model of an insurance company which holds a risky portfolio and wants to maximize payout to investors. The physics-engineering application is concerned with understanding the dynamics of phase boundaries which occur for example in chemical reactions. Both the financial and the physics-engineering models are the simplest possible prototypes. Nevertheless, sophisticated mathematical techniques are needed to understand their basic properties.
