The general area of stochastic PDEs is
interesting to mathematicians because it contains an enormous number
of challenging open problems. There is also a great deal of interest
in this topic because it has deep applications in disciplines that
range from applied mathematics, statistical mechanics, and theoretical
physics, to theoretical neuroscience, theory of complex chemical
reactions [including polymer science], fluid dynamics, and
mathematical finance.

The stochastic PDEs that are studied in this book are similar to
the familiar PDE for heat in a thin rod, but with the additional
restriction that the external forcing density is a two-parameter
stochastic process, or what is more commonly the case, the forcing is
a “random noise,” also known as a “generalized
random field.” At several points in the lectures, there are
examples that highlight the phenomenon that stochastic PDEs are not a
subset of PDEs. In fact, the introduction of noise in some partial
differential equations can bring about not a small perturbation, but
truly fundamental changes to the system that the underlying PDE is
attempting to describe.

The topics covered include a brief introduction to the stochastic
heat equation, structure theory for the linear stochastic heat
equation, and an in-depth look at intermittency properties of the
solution to semilinear stochastic heat equations. Specific topics
include stochastic integrals à la Norbert Wiener, an
infinite-dimensional Itô-type stochastic integral, an example of a
parabolic Anderson model, and intermittency fronts.

There are many possible approaches to stochastic PDEs. The
selection of topics and techniques presented here are informed by the
guiding example of the stochastic heat equation.

Readership

Graduate students and research mathematicians interested in
stochastic PDEs.