Percolation and Geometry

Lecture:

Monday, 12-14, M5
Thursday , 12-14, M5

First lecture on Monday, 12 October, 2026, 12:00-14:00

Felipe Espinosa Vergara    Pavlos Zoubouloglou

The main focus of this course will be Matrkov chains, one of the most fundamental classes of stochastic processes, with important theoretical as well as applied aspects.

Markov chains .  We will begin with the basics of discrete-time Markov chains on general state spaces. The concepts of recurrence, transience, irreducibility, invariant distributions, and ergodic theorems will be developed and proved, with numerous examples. We will also explore the connection between Markov chains and martingales, including an important application to the solution of the Dirichlet problem, which provides a probabilistic representation of solutions to PDEs.

Stationary processes and ergodic theory . We will then study stationary processes and develop the corresponding ergodic theory, subsequently establishing the connection with Markov chains. This will culminate in martingale central limit theorems and their application to central limit theorems for additive functionals of Markov chains.

Brownian motion . We will introduce Brownian motion, study salient properties of its sample paths, develop the theory of stochastic integrals, and derive Itô's formula w.r.t. Brownian motion.

Large deviations .  Finally, we will study large deviations, a theory concerned with the asymptotic behavior of small probabilities. The previously developed theory of Markov chains and Brownian motion will play an important role in this part of the course.

This lecture in the course overview.