Lecturer: Gábor Pete
Contact info: here
Lectures: Wed 09:15-11:15, 310/A. No class on Jan 18 and Feb 8. Some extra classes Wed 15:15-17:00.
Grading: There will be a written exam on Apr 3, Monday, 10-12, where proofs and some exercises may be asked. This list of theorems and this problem list will be helpful.
Lecture 1 (Jan 11): Multidimensional Gaussian distribution. A basic Gaussian tail estimate. Lévy's construction of Brownian motion. Scalings and time-inversion. Hewitt-Savage 0-1 law. BM is 1/2-\eps Hölder-continuous, but nowhere differentiable.
Lecture 2 (Jan 25): Kolmogorov extension theorem. Strong Markov property. The zero set is closed without isolated points. Reflection principle, M_t = |B_t| in distribution for fixed t. Statement of Lévy's theorem that M_t-B_t=|B_t| in distribution as processes. Uniqueness of the max place, with a gap in the proof. Definition of Hausdorff and Minkowski dimension.
Lecture 3 (Feb 1): Hausdorff dimension of the zero set is 1/2 (using the Mass Distribution Principle), that of the graph is 3/2 (here, only the upper bound). First ArcSine Law. Applications of BM, without proofs: Donsker universality, shape of random trees (Aldous' Continuum Random Tree).
Lecture 4 (Feb 15): Some ideas in Donsker's invariance principle: Skorokhod's general representation theorem; Skorokhod's embedding for the special case of a two-valued random variable. Hungarian embedding, application to the cover time of random walk on the torus (just a cultural remark). Definition and conformal invariance of harmonic measure for two-dim BM, harmonicity of \E_x[f(B_{\tau})]. Harmonicity for discrete time Markov chains.
Lecture 5 (Feb 22): Lévy processes, infinitely divisible distributions, \alpha-stable Lévy processes. Lévy-Khinchine formula. Five examples: 1) 1-dim BM is 2-stable; 2) homogeneous Poisson process is Lévy, but not stable; 3) compound Poisson process (no details); 4) a \mapsto \tau_a for 1-dim BM is 1/2-stable; 5) a \mapsto B^2(\tau^1_a) for the 2-dim BM (B^1,B^2) is 1-stable; Cauchy distribution.
Lecture 6 (Feb 29 am): Infinitesimal generator for continuous time Markov chains and for BM. Stationary and reversible measures for discrete and continuous time random walks.
Lecture 7 (Feb 29 pm): Convergence to stationarity in total variation distance for finite Markov chains, via coupling. A much better coupling for RW on the hypercube: coupon collector.
Lecture 8 (March 8 am): Some specific examples. The exact mixing time of RW on the hypercube.
Lecture 9 (March 8 pm): Pólya's theorem for recurrence vs transience on Z^d, using the Azuma-Hoeffding large deviation inequality. The spectral radius (exponential rate of decay for the return probability) on the d-regular tree.
Lecture 10 (March 22): Spectrum of Markov operator in the finite and infinite settings. Spectral gap implies fast mixing. Expander graphs.
Lecture 11 (March 29): Amenability and the Kesten-Cheeger theorem (proving only the easy direction). The notion of ergodic and mixing probability measure preserving transformations, with examples (rotations on the circle, doubling map, Gauss map, Bernoulli shifts). Birkhoff's almost sure ergodic theorem. For the proof, I followed these notes, except that I considered ergodic systems only, but the proof is the same in the more general case. Basically the same proof can be found in Durrett's book.
Consultation (March 29, pm): Answering any questions you may have.
Rick Durrett. Probability: theory and examples. 4th edition. Cambridge University Press, 2010. https://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf.
David Levin, Yuval Peres, Elizabeth Wilmer. Markov chains and mixing times. American Mathematical Society, 2008. http://pages.uoregon.edu/dlevin/MARKOV/.
Russ Lyons with Yuval Peres. Probability on trees and networks. Book in preparation, to appear at Cambridge University Press. http://mypage.iu.edu/%7Erdlyons/prbtree/prbtree.html
Peter Mörters and Yuval Peres. Brownian motion. Cambridge University Press, 2010. http://people.bath.ac.uk/maspm/book.pdf
Gábor Pete. Probability and geometry on groups. Book in preparation. PGG.pdf