Multiscale Analysis for Stochastic Dynamics
This course presents a rigorous mathematical foundation and modern analytical techniques for multiscale stochastic dynamics. Multiscale phenomena are ubiquitous in the real world, where slow macroscopic behavior emerges from fast microscopic fluctuations.
The primary objective is to equip students with theoretical tools to rigorously reduce dimensions, separate time/space scales, and approximate complex stochastic systems. Core topics include Averaging Principles and Homogenization for Multiscale Systems. Applications spanning optimal control, statistical mechanics, and finance model will be explored throughout the term.
The primary objective is to equip students with theoretical tools to rigorously reduce dimensions, separate time/space scales, and approximate complex stochastic systems. Core topics include Averaging Principles and Homogenization for Multiscale Systems. Applications spanning optimal control, statistical mechanics, and finance model will be explored throughout the term.
Lecturer
Date
8th October ~ 2nd December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Thursday | 15:20 - 17:50 | A3-3-301 | ZOOM 12 | 815 762 8413 | BIMSA |
Prerequisite
Differential equations; Probability theory
Syllabus
1. Foundations
SDEs and Diffusion processes
Markov generator and Kolmogorov equations
Ergodic theory for Markov processes and invariant measures
2 Averaging Principles
Formal scale separation of the Kolmogorov backward equation
Khasminskii’s averaging theorem and strong/weak convergence of slow variables.
Frozen coefficients method
Perturbed test functions method
3. Homogenization
SDEs with rapidly oscillating coefficients
Connection to parabolic/elliptic PDEs with periodic or random coefficients.
Solvability of the Poisson equation / Cell Problem
Diffusion approximation and effective drift/diffusion terms
4. Some applications
SDEs and Diffusion processes
Markov generator and Kolmogorov equations
Ergodic theory for Markov processes and invariant measures
2 Averaging Principles
Formal scale separation of the Kolmogorov backward equation
Khasminskii’s averaging theorem and strong/weak convergence of slow variables.
Frozen coefficients method
Perturbed test functions method
3. Homogenization
SDEs with rapidly oscillating coefficients
Connection to parabolic/elliptic PDEs with periodic or random coefficients.
Solvability of the Poisson equation / Cell Problem
Diffusion approximation and effective drift/diffusion terms
4. Some applications
Reference
G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, New York, 2008.
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus. Springer, 1991.
W. E., T. Li and E. Vanden Eijnden, Applied Stochastic Analysis, AMS, 2021.
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus. Springer, 1991.
W. E., T. Li and E. Vanden Eijnden, Applied Stochastic Analysis, AMS, 2021.
Audience
Advanced Undergraduate
, Graduate
Video Public
No
Notes Public
Yes
Language
Chinese