Kolmogorov nonlinear diffusion equation

Abstract

The Kolmogorov nonlinear diffusion equation, u_t=\frac{1}{2}u_{xx}+f(u), was first investigated by Kolmogorov, Petrovsky, and Piscounov in their celebrated paper in 1937. After a long pause, there has been renewed interest in this equation over the past decade. In extending this equation to more general settings, various tools from both analysis and probability theory have been applied, including phase plane analysis, the maximum principle, Brownian motion estimates, and the Feynman-Kac formula. It is the intent of these notes to summarize much of the work done on the Kolmogorov equation to date, while emphasizing the interaction between analytic and probabilistic techniques. These notes are to a large extent based on a course given by Don Aronson and myself last spring at the University of Minnesota in Minneapolis. (Laboratoire de Probabilités, Université de Paris VI, Spring 1982)