Partial Differential Equations and Statistical Mechanics

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

PI: Joseph G. Conlon, University of Michigan DMS-0138519 ABSTRACT This project is concerned with elliptic and parabolic partial differential equations and their applications in statistical mechanics. The first part of the proposal is concerned with elliptic and parabolic equations in divergence form where the coefficients are random variables. The author intends to study the regularity properties of the expectation value of the Green's function, and the rate of convergence of the solution of the equation to the solution of the homogenized equation. The second part of the proposal is concerned with the application of ideas from the theory of divergence form parabolic equations with random coefficients to Euclidean field theory. The relationship between elliptic and parabolic equations with random coefficients and Euclidean field theory was discovered recently by Helffer and Sjostrand. It was then more fully developed by Naddaf-Spencer. The author intends to apply some ideas he and Naddaf have developed for random elliptic equations to the Euclidean field theory situation. In the third part of the proposal the author proposes to study nondivergence form elliptic equations with random coefficients, in particular an equation corresponding to Brownian motion with a random drift. It has been shown by Sinai that in one dimension diffusion with random drift is strongly subdiffusive for large time. It has been conjectured that in dimension larger than 2 the scaling limit of diffusion with random drift is Brownian motion. The author has found a connection between this conjecture and certain combinatorial problems concerning the existence of cycles in graphs. He plans to continue his program to solve these combinatorial problems. The final part of the proposal is concerned with uniformly elliptic equations with deterministic coefficients. The coefficients can oscillate arbitrarily rapidly however. The author plans to continue his work to obtain estimates on the underlying diffusion which are independent of the degree of oscillation of the coefficients. This part of the proposal is related to the previous parts since random equations have coefficients which are rapidly oscillating. The goal of the proposal is to understand properties of the solution to a partial differential equation when the only information one has is that the coefficients of the equation are bounded in some way. Within this goal there are two sub-themes: (a) understanding worst possible behavior, (b) understanding "on average" behavior -given our knowledge of the coefficients. The sub-theme (a) is the subject of the final part of the proposal. It is intimately related to problems of stochastic control theory. Examples of stochastic control theory abound in the world of engineering and finance. To take a financial example, consider the problem of valuing a stock option. The classic work of Black and Scholes shows that the value of the option depends only on the stock volatility. For a stock with constant volatility they have a formula for the value of the option in terms of the volatility. The formula is the solution to a partial differential equation in which the volatility is a coefficient. (a) is therefore related to the problem of estimating worst possible scenarios for option values when one can only assume some bounds on stock volatility. The sub-theme (b) has similar applications to (a). The most exciting of these to this author is that it offers a way of beginning to understand the problem of turbulence in fluids. Turbulence is roughly speaking the onset of random behavior in the velocity of a fluid. It is well known that a fluid will undergo turbulent behavior when subject to a sufficiently large disturbance. The mathematical problem of understanding turbulence is well defined. One simply needs to understand the solutions of a partial differential equation known as the Navier-Stokes equation. To date there is not even the beginnings of an understanding how to derive turbulence in a mathematically rigorous way out of the Navier-Stokes equation. The reason is that the fluid velocity is a coefficient of the equation. In the turbulent regime therefore the Navier-Stokes equation is like a partial differential equation with a random coefficient. The sub-theme (b) is then concerned with the "typical" behavior of the fluid velocity in this situation.