Slides & board photos, material covered

Lecture notes: (WARNING: These can change day to day.)

Slides & board photos & synopsis of material covered:

Lecture 1, March 31: Introduction to the course. Review of ergodic theory. Measure-preserving transformations, examples. Ergodic theorems of von Neumann and Birkhoff. The Spatial Ergodic Theorem and its proof via Wiener's covering lemma and the ensuing maximal inequality. (slides)
Lecture 2, April 2: The Spatial Ergodic Theorem, details of the proof with emphasis on infinite measure spaces. Integrability of the maximal function. Wiener's Dominated Ergodic Theorem. Banach's observation that finiteness of maximum function suffices. (photos)
Lecture 3, April 4: Invariant and almost invariant sets. Khinchin's interpretation of limit function in Birkhoff's theorem. Ergodicity and its characterization by absence of non-constant invariant measurable functions. Examples: i.i.d. shifts, irrational rotations, Boole's transform. Non-examples as well. (photos)
Lecture 4, April 7: Further necessary and sufficient conditions for ergodicity. Absence of non-simple eigenfunction of the induced operator. Ergodicity as an irreducibility property: mutual singularity and extremality. Poincaré Recurrence Theorem. (photos)
Lecture 5, April 9: Khinchin and Kac Recurrence Theorems. The consequences for the latter in statistical mechanics. Proof by way of Kakutani skyscraper. Applications of Poincaré Recurrence Theorem to induced map. Recurrence in infinite measure spaces; wandering sets. (photos)
Lecture 6, April 11: Halmos' recurrence theorem: equivalence of recurrence, conservativity, incompressibility and infinite recurrence. Hopf decomposition for regular bimesurable bijections. Stephanov-Hopf Ratio Ergodic Theorem. Key lemma from Kamae and Keane's paper. (photos)
Lecture 7, April 14: Proof of Stephanov-Hopf Ratio Ergodic Theorem. Weak and strong mixing and relation to recurrence, ergodicity and (for stationary random variables) tail triviality. (photos)
Lecture 8, April 15: Characterization of tail triviality. Example showing that tail triviality is stronger than strong mixing. Irrational rotation of the unit circle is not strongly mixing. Koopman-von Neumann lemma linking vanishing Cesaro averages to convergence to zero except for zero density set. Convergence in density. (photos)
Lecture 9, April 16: Characterization of weak mixing transformations. Spectral characterization by absence of eigenvalues different from 1. Subadditive sequences and Fekete lemma. (photos)
Lecture 10, April 18: Applications of Fekete's lemma to counting self-avoiding paths, expected range of the random walk and spectral radius defined via operator norms. Liggett's Subadditive Ergodic Theorem, proof of key lemmas. The final touches of the proof to be done next time. (photos)
Lecture 11, April 21: The proof of the Subadditive Ergodic Theorem completed. Applications to range of random walk, longest increasing subsequence and first-passage percolation metric, including the statement of the shape theorem for balls in this metric. (photos)
Lecture 12, April 22: Markov chains defined by Markov property. Transition probability and construction of Markov chain with given initial distribution and transition probabilities. Time-homogeneous Markov chain. Examples: Two-state chain, Ehrenfest chain, random walk as the sum of i.i.d. random variables, random walks on graphs, exclusion process. (photos)
Lecture 13, April 23: Invariant/stationary measures of a Markov chain. Existence for finite state spaces by Perron-Frobenius Theorem. Existence for compact state spaces with Feller transition probabilities. Examples: Ehrenfest chain and card shuffling. Multiple infinite invariant measures for biased simple random walk. A chain with no nontrivial invariant measure. (photos)
Lecture 14, April 25: Irreducibility. Bi-infinite extension of stationary Markov chains. Reversed chain and reversible measures. Examples: Birth-death chain, random walks on graphs, biased random walk. The Kolmogorov cycle conditions as a necessary and sufficient condition for reversibility. (photos)
Lecture 15, April 28: Stationary Markov chains are Markov shifts. Characterization of reversibility by self-adjointness of transition operator. Markov and strong Markov property. Recurrent state and characterization of recurrence by diverging expected number of returns. (photos)
Lecture 16, April 29: Existence of invariant measure assuming existence of a recurrent state. Uniqueness modulo scaling under irreducibility. Relation between finiteness of the measure and positive recurrence. Density of returns and ratio convergence theorems. (photos)
Lecture 17, April 30: Renewal chain loses invariant measure (by tuning parameters) exactly when it becomes transient. Cezaro convergence for distributions of Markov chain. Aperiodicity. Irreducible, aperiodic and positive recurrent Markov chains converge to their stationary distribution. Proof by coupling. (photos)
Lecture 18, May 2: Mixing time for Markov chains. Recursive bound on TV distance. Example of lazy random walk on a ring. Aldous-Diaconis example of card shuffle and cutoff phenomenon occurring there. Analysis using coupon-collector problem computations. (photos)
Lecture 19, May 5: Pólya's theorem: The simple symmetric random walk is recurrent in dimensions 1 and 2 and transient otherwise. Extension of recurrence to random walks satisfying WLLN (dimension 1) and CLT (dimension 1 and 2). "Truly d-dimensional random walks are transient when d is 3 or higher. Harmonic functions for a Markov chain. Recurrence implies that bounded and/or non-negative harmonic functions are constant. All bounded harmonic functions are constant for simple random walk in any dimension. (photos)
Lecture 20, May 6: A non-constant bounded harmonic function for random walk in degree-3 tree. One-to-one relation between bounded harmonic functions and shift invariant bounded functions of Markov chain path. Choquet-Deny theorem and characterization of non-negative harmonic functions on ℤ^d as mixtures of harmonic exponentials. (photos)
Lecture 21, May 7: Examples demonstrating the power of the Choquet-Deny theorem: Absence of positive harmonic function under vanishing first moment of step distribution, existence under exponential integrability and non-zero first moment. Harmonic analysis of Markov chains: Solution of Dirichlet problem. Characterization of transient chains by existence of a non-constant superharmonic non-negative function. Riesz decomposition of superharmonic non-negative function into a potential and a harmonic function. (photos)
Lecture 22, May 9: Martin boundary representation of harmonic functions. Examples: biased random walk and random walk on a tree. Reversible chain and connection to electrostatic theory. Notions of Dirichlet energy and effective conductance. Minimizer by solving Dirichlet problem. Thomson problem and definition of effective resistance. (photos)
Lecture 23, May 12: Effective conductance is the value of the flow to achieve unit voltage difference. Effective resistance is the reciprocal value of effective conductance. Ohm's law for the minimizers. Relation of effective resistance and Green function. Network reduction, parallel and series laws, star-triangle transformation, shortening. Effective resistance and conductance to infinity and its use to characterize transience. The Nash-William's criterion for recurrence. (photos)
Lecture 24, May 13: Using the Nash-William's criterion to prove recurrence of simple random walk in dimensions 1 and 2. Criterion for transience: Existence of unit flow to infinity of finite energy. Terry Lyons' random path method, proof of transience of SRW in dimensions 3 and higher. The Rayleigh monotonicity principle. Applications to random conductance model. Markov chains in continuous time: motivating examples. The transition function and Q-matrix. (A photo of this board is missing.) (photos)
Lecture 25, May 14: Continuous-time Markov chain theory: From transition function to Q-matrix. Instantaneous, stable, absorbing and defective states. Blackwell's example of a countable-state continuous-time Markov chain with all states instantaneous. (photos)
Lecture 26, May 28: Recap of topics in continuous-time Markov chain theory covered so far. Absence of instantaneous and defective states in finite-state Markov chains; one-to-one correspondence between transition functions and Q-matrices there. Existence of transition function for a Q-matrix with bounded diagonal entries. General case: minimal positive solution to Backward Kolmogorov Equation. Its representation using a discrete-time Markov chain and a Poisson process. Equivalence of stochasticity and non-explosion. (photos)
Lecture 27, May 30: A criterion for non-explosivity and its validity under bounded rates and/or recurrence of the underlying discrete-time chain. Definition of continuous-time Markov chain. The Markov chain gives a transition function but (by Blackwell's example) not always the other way round. A direct way from the Markov chain to Q-matrix: absence of instantaneous and defective states and non-explosivity. Key tool: Strong Markov property.(photos)
Lecture 28, June 2: Proof of Theorem showing that Markov chain gives well defined transition function and Q-matrix which reproduces the chain back. Transition function determines the law of the chain uniquely. The Forward Kolmogorov equation. Semigroup formulation. Stationary measures. Partial characterization using Q-matrix. (photos)
Lecture 29, June 3: Markov processes on general state spaces. The setting and definition, including the Markov and Feller property. Every Markov process defines a strongly continuous probability semigroup. Example of semigroups: uniform shift, convolution with normal or Cauchy law. Every probability semigroup (with good approximation of unity in locally compact spaces) determines a unique Markov process. Proof that the semigroups identifies the finite-dimensional distributions.(photos)
Lecture 30, June 4: From semigroup to Markov process: a.s. existence of one-sided limits via supermartingales. Example of the standard Brownian motion and its characterization by distribution of increments and sample-path continuity. Infinitesimal generator and boundedness of the resolvent operator. Reconstruction of the semigroup from the generator. (photos)