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.
讲师
日期
2026年10月08日 至 12月02日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周三,周四 | 15:20 - 17:50 | A3-3-301 | ZOOM 12 | 815 762 8413 | BIMSA |
修课要求
Differential equations; Probability theory
课程大纲
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
参考资料
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.
听众
Advanced Undergraduate
, Graduate
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笔记公开
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语言
中文