# Math 110.757 - Topics in Stochastic Dynamical Systems

Instructor:     Fei Lu
Class meets:   TTh, 9-10:15, Garland 97
Office Hours: TTh 10:15--11:15,   Krieger 301
Webpage:          http://www.math.jhu.edu/~feilu/19Spring/StoDS/stoDS19.html
Email:             feilu##   ( ## = @math.jhu.edu)

Textbook:  Rafail Khasminskii: Stochastic stability of Differential equations

•   Ludwig Arnold: Random Dynamical systems
•   GA Pavliotis; AM Stuart: Mutliscale methods: averaging and homogenization
•   Kody Law;  Andrew Stuart; Konstantinos Zygalakis. Data Assimilation: A Mathematical Introduction

Course plan (tentative): The course will present a introduction to stochastic dynamical systems and some applications in model reduction and data assimilation. The main focus will be on stability and ergodicity of stochastic dynamical systems, including stochastic differential equations driven by white noise and fractional noise, and their numerical approximations. We will then discuss applications of stochastic dynamical systems, related topics include inference, model reduction and data assimilation. The course is open to majors in math, applied math, statistics, engineering. I will try to accommodate the backgrounds of students.

Prerequisite: analysis (real analysis, functional analysis); probability; differential equations (preferably stochastic differential equations).

Grading: Grade will be based on homework assignments and presentations. There is no exam.
Stochastic Grading: Your grade will be based on presentation of homework exercises. Each week, there will be 1-3 exercises assigned. One student will be randomly picked to present his/her solution to a randomly picked excercise in class in 3-5 minutes. The audience can ask questions to help identifying the gaps in the presenter's proof. The presenter gets 2-3 points, the audience who helped identifying a gap gets 1 point. We will randomly pick one student to type up the solution, who will get 1 point. In the end, grade will be assigned based on statistics. Above mean: A; [mean - 2 std, mean]: B; else, C.

Tentative schedule:

week Topics Sections Other
1/29, 1/31
Chp1

2/5, 2/7
Chp2
2/12, 2/14
Chp3
2/19, 2/21
Chp3
2/26, 2/28
Chp4
3/5, 3/7
Chp4
3/12, 3/14 Stochastic Approximation: Chp7.5
Robbins-Monro51: A stochastic approximation method
Bottou et al18: Optimization Methods for Large-Scale Machine Learning
Cucker-Smale01: On the mathematical foundations of learning

3/19, 3/21 spring break
3/26, 3/28 SDE with fBm Hairer05: Ergodicity of stochastic differential equations driven by fractional Brownian motion
HO07:  Ergodic theory for SDEs with extrinsic memory

4/2, 4/4
Class on 4/2 cancelled; to be made-up on May 7 and 9th

4/9, 4/11 measure preserving DS
Arnold: appendix

4/16, 4/18 Random and stochastic DS
Arnold: chapter 1-2, HO07

4/23, 4/25 Mori-Zwanzig formalism
Chorin-Hald (2013) Chapter 9; Slides of Li (2019) Hudson-Li (2018)

4/30, 5/2 Gradient systems and Hamiltonian systems

5/7, 5/9 Inference of SDE