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. For example,

  • 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):

    1. 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).
    2. 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.
    3. Falconer, Fractal geometry: mathematical foundations and applications, (Ch. 2-4, a survey of concepts needed for analyzing random fractals).
    4. Peres 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).
    5. 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.

    Requirements: 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.

    Instructor: Amir Dembo, M 2:00-3:00, Stat. 129 or e-mail to set an appointment.

    Official course lecture notes (PDF) .

    See also seminar for current activity in related areas.