Stochastic Partial Differential Equations with a Linear Potential

University Of Rochester, Rochester NY

Investigators

Abstract

Many partial differential equations describing the physical world, such as the Schrodinger equation, involve a potential term. We propose to study stochastic partial differential equations with a Gaussian random potential. There are many examples of such equations, and usually they are treated on a case-by-case basis. On the other hand, in many cases the solution can be expanded in terms of multiple stochastic integrals with respect to the noise. There are many tools available to study such equations, including Wiener space analysis and the Feynman-Kac formula. We intend to use these tools to study qualitative properties of solutions, such as asymptotic growth, and the critical parameters for existence of solutions. For certain parameter values, we expect solutions taking values in the space of Schwartz distributions. Of course, this is a broad program, and many such equations have been considered before. However, much less is known for the case of correlated Gaussian noise than for white noise. Here, we are referring to specific phenomena, which might occur only for certain correlation functions of the Gaussian noise. Also, the stochastic heat equation has received much more attention than other cases. In the physical sciences, engineering, and increasingly in biology, the most useful tool is a partial differential equation. Solving the equation gives us a quantitative understanding of the physical situation, allowing us to make predictions and to control the system. However, in the real world, all systems are affected by random noise, and our models must also include randomness. The field of stochastic partial differential equations attempts to address this situation. This field is much newer than partial differential equations, and our mathematical understanding is much less. This proposal focuses on equations with a potential term. Such equations model many natural phenomena. In quantum mechanics, potential terms represent the interaction between particles and forces acting on the system. Potential terms can also arise in the equations for the spread of epidemics, and in other situations in population biology. We will study the effect of a random potential. The goal is to increase our understanding of the mathematical tools, so that scientists and engineers can use them more skillfully.