Topics in Probability (MATH232, same as STAT350, Spring 2003)
The central theme of this course shall be multiscale occupation analysis:
Favourite points, cover times and fractals. We shall explore the fractal
structure of random sets associated with occupation measures of the most
fundamental stochastic processes: random walk, Brownian motion and
stable processes. A common theme is the tree like correlation structure
of excursion counts around different centers, which makes a
multi-scale refinement of the second moment method effective.
After a short review of recent advances in this topic accompanied
with open problems for future research, we focus on key methods
of possible independent interest, demonstrated by applications.
Concentration via multiscale second moment computation.
Packing and Hausdorff dimensions for discrete limsup random fractals.
Concentration of cover time for Markov chains (Matthews method).
Strong approximation, the KMT construction.
Linear operators and Ciesielski-Taylor type identities.
The central theme of the course is taken from recent papers such as:
"Thick Points for Planar Brownian Motion and the Erd\"os-Taylor
Conjecture on Random Walk" (Acta Math. 186 (2001), pp. 239-270),
"Cover Times for Brownian Motion and Random Walks in Two Dimensions"
"Brownian Motion on compact manifolds: cover time and late points",
"Late points for random walks in two dimensions",
and the references therein. See also
red color for thick points and
yellow color for late points (produced by Raissa D'Souza).
Supplementary Texts (updated as we progress):
Aldous and Fill
Reversible Markov chains and random walks on graphs (see
Matthew's method in Ch. 2.6 and examples of cover time problems
in Ch. 6).
Revesz, Random walk in random and non-random
environments, has many open problems and conjectures about
favorite points and cover times for simple random walk.
Falconer, Fractal geometry: mathematical foundations and applications,
(Ch. 2-4, a survey of concepts needed for analyzing random fractals).
An invitation to sample path of Brownian motion
(which deals with fractal geometry of simpler random sets related
to the sample path of the Brownian motion).
Lawler, Intrsections of random walks (Ch.1 only, where potential theory
estimates are adapted from Brownian motion to random walk).
Prerequisites: Some familiarity with probability
theory and stochastic processes at the level of Stat310, or
at the level of Math136, with instructors consent.
Each participant shall prepare a LaTeX version of one week's
lectures to be posted on this site the following week.
Meeting: McCullough 122, TTh 11:00-12:15.
First meeting, April 1st.
M 2:00-3:00, Stat. 129 or e-mail
to set an appointment.
seminar for current activity in related areas.