2024/2025
Ergodicity and Mixing for Markov Processes
Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Language:
English
ECTS credits:
6
Course Syllabus
Abstract
Last decades witness fast penetrating of Probability to other fields of mathematics, first of all - to differential equations, PDEs and mathematical physics. Probability starts to enter geometry, and has rich and involved relation with the algebraic number theory. The goal of this course is to present sections of Probability, motivated by differential equations, PDEs and mathematical physics in order to justify physical thesis that "complicated systems during their evolution converge to statistical equilibria". This convergence is called "the mixing", and the thesis above is, for example, one of postulates of the theory of turbulence. (For the situation which appears there, it still is not justified due to extreme complexity of the 3d Navier-Stokes system, which governs motion of fluids and gases).
Course Contents
- Some basics from probability and measure theory.
- Independence, conditional expectation and conditional probability.
- Spaces of measures and metrics on them.
- Markov processes in metric spaces: Transition function and Markov semigroups; Construction of a Markov process from a transition function.
- Markov property. Stopping times and strong Markov property.
- Markov processes, defined by stochastic equations.
- Doeblin condition and Doeblin mixing theorem.
- Harris’ mixing theorem.
- Law of large numbers.
- Some examples from physics.
Interim Assessment
- 2024/2025 4th module0, 1 ∗ 𝐻 + 0, 4 ∗ 𝑀 + 0, 5 ∗ 𝐹 where 𝐻 is for the home work grade, 𝑀 is for the midterm grade and 𝐹 is for the final exam grade.
Bibliography
Recommended Core Bibliography
- Вероятность. Кн.1: Элементарная теория вероятностей. Математические основания. Предельные теоремы, Ширяев, А. Н., 2017
- Вероятность. Кн.2: Суммы и последовательности случайных величин - стационарные, мартингалы, марковские цепи, Ширяев, А. Н., 2017
Recommended Additional Bibliography
- Случайные процессы : учебное пособие, Шалимов, А. С., 2024