MATH-487 / 6 credits

Teacher: Li-Hairer Xue-Mei

Language: English

Summary

This course offers an introduction to topics in stochastic analysis, oriented about theory of multi-scale stochastic dynamics. We shall learn the fundamental ideas, relevant techniques, and in general improve our knowledge of stochastic processes. We touch also trends in current research.

Content

We introduce two-scale systems with slow and fast variables. These variables evolve interactively, but at very different speed. From the standpoint of the slow variable, the fast variables are not tractable.  It feels the influence of the fast variable. Multi-scale theory is concerned with iddentifying the effect of the the fast on the slow variable. For two scale interactive slow /fast system of (stochastic) differential equations, we seek an autonomous equation whose solutions approximate the slow variables when the `separation of scale' parameter is large. This theory is strongly linked with ergodicity. Prime examples are Markov processes and solutions of stochastic differential equations. We hope to give an overview of the classical results and touch on recent development and modern techniques.

Motivating models include the evolution of celestial body orbits in an approximate random Hamiltonian system and the approximation of Brownian motions using stochastic processes with a velocity field an Ornstein-Uhlenbeck process, and climat versus weather.

Keywords

Stationary process, ergodicity, Birkhoff's ergodic theorem,  Markov processes, invariant measures and ergodicity of Markov processes, Functional large of large numbers for Markov processes, Functional central limit theorems, quantitative theory, and martingales. Special processes such as Ornstein-Uhlenbeck processes and some models  involving stochastic differential equations.

Required courses

Good knowledge of the following are required: Analysis, Probability, Stochastic Processes, Measure and Integration, knowledge of differential equations (ODE /PDE), Metric spaces and functional analysis.

Indicative foundational EPFL courses are:

Measure ans Integration (Math 303), Probability Theory (Math 432), Stochastic Processes (Math 332), Martingales et mouvement brownien (MATH-330), Stochastic Calculus (Math 431)

Learning Outcomes

By the end of the course, the student must be able to:

• Apply their understanding to develop proofs of unfamiliar results
• Apply these concepts and results to tackle a range of problems, including previously unseen ones
• Demonstrate additional competence i nthe subject through the study of more advanced material
• Explain thier knowledge of the area in a concise, accurate and coherent manner
• Demonstrate understanding of the concepts and results from the syllabus includign the proofs of a variey of results

Teaching methods

Lectures and Exercise classes

Expected student activities

Attend lectures, problem classes, do exercises and extra reading

Oral

Supervision

 Office hours No Assistants Yes

Bibliography

--Random Perturbations of Dynamical Systems by Mark I. Freidlin , Alexander D. Wentzel, --ASYMPTOTIC ANALYSIS FOR PERIODIC STRUCTURES by A. BENSOUSSAN J.-L. LIONS G. PAPANICOLAOU, fluctuations in Markov Processes, Time Symmetry and Martingale Approximation

-- Tomasz Komorowski , Claudio Landim , Stefano Olla, Markov Processes, Characterization and Convergence

--Stewart N. Ethier, Thomas G. Kurtz Stochastic Differential Equations and Diffusion Processes
https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/24

--Continuous Martingales and Brownian Motion, Daniel Revuz, Marc Yor

--- Measure Theory, Probability, and Stochastic Processes. J-F Le Galll

--Convergence of Markov Processes

https://www.hairer.org/notes/Convergence.pdf, Martin Hairer

-- Ergodic properties of Markov processes. Martin Hairer
https://www.hairer.org/notes/Markov.pdf

In the programs

• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Fall
• Exam form: Oral (winter session)
• Subject examined: Topics in stochastic analysis
• Lecture: 3 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

Reference week

Tuesday, 10h - 12h: Lecture GRA330

Wednesday, 10h - 11h: Lecture CM1104

Friday, 13h - 15h: Exercise, TP GCD0386

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