Course Outline. "Lévy processes" are "random walks that evolve in continuous
time." Fundamental stochastic processes such as Poisson processes, Brownian
motion, and stable processes are archetypal examples of Lévy processes. And every
"natural" Markov process on \({\bf R}^n\) is "locally a Lévy process."
Those who know some mathematical finance might know about Lévy processes such as CGMY
and gamma processes, as well.
Outside mathematics, Lévy processes are known also as ``Lévy flights,'' and play
a central role in the stochastic modeling of random processes that ``have heavy-tailed
distributions.''
The goal of this course is to study Lévy processes. As it turns out, this goal provides
us with a natural context within which we can learn diverse and important topics such as point
processes, heavy-tailed distributions, Markov processes, pseudo-differential operators, ... .
Special emphasis is placed on studying concrete [i.e., important!] families of Lévy processes.
The course will cover the following topics:
Infinitely-divisible distributions;
Point processes;
Structure theory [Lévy-Itô representation];
Bochner's subordination;
The strong Markov property;
Pseudo-differential operators, the associated semigroup, potential theory;
Energy and capacity [time permitting].
Prerequisites. Basics of measure-theoretic probability at the level of Math. 6040. Text. Lecture notes will be handed out. Grading. Exercises will be assigned throughout. Registered students can choose
to either: (i) Give 1 lecture on a published paper on/around this topic; or (ii) Turn in
weekly exercises.