Analysis of Stochastic Partial Differential Equations
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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.
A co-publication of the AMS and CBMS.
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