Analysis of Stochastic Partial Differential Equations

The Michigan State University

The principle aim of these lectures is to begin with general fundamental notions such as the mathematics of stochastic partial differential equations, and end by presenting a series of recent advances and various powerful methods, as well as various problems, from mathematics and mathematical physics, surrounding the general topic of ``intermittency" in the context of various nonlinear stochastic PDEs. Some of the problems will be particularly suitable as research projects for graduate students and post-doctoral researchers.

The only pre requisites are a solid course in measure-theoretic probability, and a modest knowledge of Brownian motion and its associated stochastic calculus.

The following is a more detailed plan for the lectures.

**1. ***Introduction: overview and applications*.
We present an informal description of stochastic partial differential
equations. Concrete examples of SPDEs are drawn from science and engineering.

**2. ***Interacting diffusions.* We introduce and
study examples of families of interacting diffusions in order to discuss many
of the fundamental techniques of SPDEs in a less technical setting. Particular
attention is paid to two models of mathematical physics: The parabolic Anderson
model; and Funaki's discrete approximation to the ``random string.''

**3. ***Gaussian noise*. One of the fundamental
objects in SPDEs is ``noise.'' Here we lay rigorous foundations of Gaussian
noise [including ``white noise'' and ``colored noise''], as well as stochastic
integration \`a la Wiener, It\^o, and Walsh.

**4. ** *Linear
equations.* Linear SPDEs are introduced and studied. This is a simple
setting in which one can learn many of techniques that will be used later on in
order to analyze more complicated SPDEs. We describe various structural
properties of the solutions to linear SPDEs that highlight the effect of noise
[or lack thereof] in the behavior of the solution.

**5. ***Non-linear equations*. Non-linear
equations are introduced, and defined rigorously. General issues of existence
and uniqueness are addressed. Special attention is paid to various function
classes that arise naturally in the context of some non-linear SPDEs of
interest.

**6. ***Local behavior*. The preceding lectures
cover aspects of the ``standard theory.'' From here on, we study in greater
detail more concrete families of SPDE models. In these two lectures we continue
our analysis of non-linear SPDEs by studying various local properties of these
solutions. Typical examples of such local properties are regularity theory
[smoothness of the solution], the analysis of the local effect of noise,
comparison principles [including positivity principles], and several of their
consequences.

**7. ***Intermittency*. (multiple lectures) In
the first lecture on this module, we first introduce physical intermittency,
via examples, together with various mathematical models that attempt to
describe physical intermittency. Then we establish moment estimates that show
the existence of non-trivial moment Lyapunov exponents, thereby suggesting the
existence of ``intermittent islands'' for a class of non-linear SPDE models. In
the subsequent two lectures we describe aspects of the recently-developed description
of the geometry of the intermittency islands.