Math757: a topic course in stochastic dynamical systems
Instructor: Fei Lu
Class meets: TTh, 10:3011:45, Shriver Hall 104
Office Hours: TTh 9:3010:30, Krieger 301
Webpage:
http://www.math.jhu.edu/~feilu/20Spring/StoDS/stoDS.html
Email:
feilu## ( ## = @math.jhu.edu)
This topic course will run in a reading/discussion/projects fashion. References are attached to each topic. In addition, the following books may be helpful (To be updated).
Course plan (tentative):
This topic course will explore systems of interacting particles or agents: their dynamics, inference, and applications in machine learning. For the study of the dynamics, topics include stability and ergodicity of the systems, including firstorder gradient systems, 2ndorder Hamiltonian systems and related mean field equations and propagation of chaos. For the applications, topics include optimization algorithms such as stochastic gradient decent (SGD) and particle SGD, and sampling methods using particle systems such as Stein variational gradient decent. For the inference, we will consider the estimation of the interaction kernels as well as state estimation using data assimilation techniques. If time permits applications in network and control with be explored. This topic course will run in a reading/discussion/projects fashion.
Prerequisite: probability; differential equations (preferably stochastic differential equations).
Grading: Grade will be based on project assignments and presentations. There is no exam.
Tentative schedule (will be updated weekly):
week  Topics  References  Other 
1/28, 1/30 
Course plan overview Review ODE (Caratheodory DE)

nonLipschitz / discontinuous RHS; existence, uniqueness, global solution
LectureNotes1 
Homework 1 tex file Due on 2/11; solution 
2/4, 2/6 
Consensus and flocking Dynamics

Motsch+Tadmor14, Heterophilious dynamics enhances concensus Cucker+Smale07, Emergent behavior in flocks 

2/11, 2/13 
Consensus and flocking Dynamics

LectureNotes2 

2/18, 2/20 
Mean field

Bertozzi et al: Blowup in multidimensional aggregation equations with mildly singular interaction kernels
Reading: Simione et al: Existence of ground state of nonlocalinteraction energies
Ambrosio+Bernard: Uniqueness of signed measures solving the continuity equation for Osgood vector fields
LectureNotes3


2/25, 2/27 
stochastic systems: 1st order

basics LectureNotes4


3/3, 3/5 
stochastic systems: 1st order

Malrieu03: Convergence to equilibrium for granular media equations and their Euler schemes
Toscani+Villani00: On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds
Jabin+Wang17: mean field limit for stochastic particle systems
LectureNotes5

Homework2 tex file Due on 3/24 Extended to 3/31 Cattiaux+Guillin+Malrieu08ptrf Mattingly+Stuart+Higham02SPA

3/10, 3/12 
stochastic systems: 1st order

Implicit Euler LectureNotes6 

3/17, 3/19 
spring break

3/24, 3/26 
stochastic systems: 1st order 2nd order 
Implicit Euler (Malrieu03) LectureNotes6 Further reading: Guionnet+Zegarlinski02: lectures on LSI Bolley+Gentil+Guillin13: Uniform convergence to Equilibrium for Granular media
Bakry04 Functional Inequalities for Markov semigroups
Linear 2ndorder models LectureNote Record 

3/31, 4/2 
stochastic systems: 2nd order 
stochastic CuckerSmale model: LectureNote0331 LectureNote0402
Pedeches18: Asymptotic properties of various stochastic CuckerSmale dynamics
Cattiaux+Delebecque+Pedeches18: stochastic CuckerSmale models: old and new


4/7, 4/9 
Exponential Ergodicity of SDEs 
Mattingly+Stuart+Higham02SPA

4/14, 4/16 
Inference of the interaction function 
basics of nonparametric regression: (CS01) CuckerSmale01: On the mathematical foundations of learning Chp10 of (GKKW02) Gyorfi+Kohler+Krzyzak+Walk: A distributionfree theory of nonparametric regression 

4/21, 4/23  Inference of the interaction function

deterministic and stochastic systems LecNote 

4/28, 4/30  Inference of the interaction function: stochastic systems 
Likelihood ratio: Chp1.1.1.4 Kutoyants04 Statistic inference for ergodic diffusion processes ; Chapter 3.5 of Karatzas+Shreve: Brownian motion and stochastic calculus Concentration for Martingales: Gene+Nickl: Math Foundation of infinitedimensional statistical models; Lemma 1 of Nickl+Ray1810 LecNote 
