Qualitative Properties of Stochastic Partial Differential Equations

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This proposal deals with three topics in stochastic partial differential equations (SPDE). The first question deals with uniqueness. Unless SPDE have unique solutions, they are useless for modeling. Recent work by the proposer, L. Mytnik, and E. Perkins settled a longstanding question about uniqueness of SPDE related to the heat equation. Using these newly developed tools, the proposer will attack uniqueness questions for other types of equations, and study uniqueness among nonnegative solutions. The second question deals with traveling waves, which occur widely in physical systems. Recently the proposer, L. Mytnik, and J. Quastel have solved a problem raised by Brunet and Derrida, involving the asymptotic speed of traveling waves when the noise term is small. The proposer will would like to extend this analysis to other equations, and related particle systems. The third topic involves stochastic wave equations. The most commonly studied SPDE are variants of the heat equation, but others such as the stochastic wave equation are receiving increasing attention. With D. Geba, the proposer will like to study stochastic wave equations involving null forms. Using some function spaces introduced by Bourgain, the proposer's goal is to study short-time existence. The ideas developed should be relevant to other classes of equations. The field of SPDE is rapidly expanding. As technology moves towards the micro level, the effect of random noise becomes more and more important. Since the most important mathematical modeling tools we have are ordinary and partial differential equations, the study of SPDE is becoming essential in many applied fields. Even though SPDE is now several decades old, the area has only recently received widespread attention. There is still a need for pioneering work to establish a toolbox for the area. This proposal deals with three types of problems in SPDE. The proposer believes that through the study of such specific examples is the right way to generate methods and further our understanding, for both theory and applications. In summary, the proposer believes that SPDE will play an essential role in future applications of mathematics. He wishes to develop the theory and help to train graduate students so that this area can fulfill its potential.