This book aims to give a user friendly tutorial of an interdisciplinary
research topic (fronts or interfaces in random media) to senior
undergraduates and beginning grad uate students with basic knowledge of
partial differential equations (PDE) and prob ability. The approach
taken is semiformal, using elementary methods to introduce ideas and
motivate results as much as possible, then outlining how to pursue rigor
ous theorems, with details to be found in the references section. Since
the topic concerns both differential equations and probability, and
proba bility is traditionally a quite technical subject with a heavy
measure theoretic com ponent, the book strives to develop a simplistic
approach so that students can grasp the essentials of fronts and random
media and their applications in a self contained tutorial. The book
introduces three fundamental PDEs (the Burgers equation, Hamilton-
Jacobi equations, and reaction-diffusion equations), analysis of their
formulas and front solutions, and related stochastic processes. It
builds up tools gradually, so that students are brought to the frontiers
of research at a steady pace. A moderate number of exercises are
provided to consolidate the concepts and ideas. The main methods are
representation formulas of solutions, Laplace meth ods, homogenization,
ergodic theory, central limit theorems, large deviation princi ples,
variational principles, maximum principles, and Harnack inequalities,
among others. These methods are normally covered in separate books on
either differential equations or probability. It is my hope that this
tutorial will help to illustrate how to combine these tools in solving
concrete problems.
