XXXIII St.Flour Probability Summer School, July 2003

Official course lecture notes (PDF) .

The central theme of this course shall be multiscale occupation analysis: Favorite 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 (Chapter 1), we focus on key methods of possible independent interest, demonstrated by applications. For example,

  • Concentration of cover time for Markov chains (Chapter 2).
  • Properties of discrete limsup random fractals (Chapter 3).
  • Multiscale truncated second moment computation (Chapter 4).
  • Strong approximation, the KMT construction (Chapter 5).
  • Moment computations and Ciesielski-Taylor identities (Chapter 6).

    Planned lectures:

  • Overview (Section 1.1).
  • Expected cover time for Markov chains (Section 2.1).
  • Brownian cover time, 3-dimensional torus (Section 2.3).
  • Localization and thick times for Brownian motion (Section 3.3; Read 3.1 before).
  • Favorite and perfect leaves for SRW on regular trees (Section 4.2).
  • Multiscale truncated second moment computation (Section 4.3).
  • Brownian motion to planar SRW via KMT construction (Section 5.1).
  • From Regular trees to planar Brownian motion (Section 5.3).
  • Discrete limsup random fractals (Section 3.2).
  • Supplements as needed or overview from Chapters 1 and 6.

    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" (Ann. Math. to appear (2003)),
  • "Brownian Motion on compact manifolds: cover time and late points", (Elec. J. Probab. (2003)),
  • "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 :

    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. Le Gall, 1990 St.Flour lecture notes (more on path properties of Brownian motion than we do here).
    6. Peres, 1997 St.Flour lecture notes (probability on trees is closely related to parts of Ch. 3-4 of my notes).
    7. Revuz and Yor, Continuous martingales and Brownian motion (much of Ch. 6 of the course is there).