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MATH 778: Stochastic
Processes - Random Walk on the Integer Lattice (Spring 2004)
Instructor:
Gregory Lawler
Meeting
Time & Room
In 1964, Frank Spitzer published the classic book on random
walks, Principles of Random Walk. We will read this book (and in places
update it). This will be the start of a long range project of mine to
produce a working person's guide to random walk. We will restrict ourselves
to walks on the integer lattice (both discrete and continuous time). There
is plenty to consider in this simple setting!
We will also consider questions about strong convergence of random walk
to Brownian motion including the KMT approximation, Beurling estimates,
and their use in approximating rates of convergence on very rough domains
(in two dimensions).
As pointed out by Spitzer, random walk on the integer lattice is essentially
the same subject as discrete potential theory and this course could be
of interest to people in differential equations and combinatorics. Convergence
rates of random walks to Brownian motion are very closely related to rates
of convergence of finite difference approximations to elliptic equations.
Prerequisites will be the equivalent of Math 671, in particular familiarity
with discrete martingales and some knowledge of Brownian motion.
Last modified:
October 23, 2003
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