Course No.：216032Y
Period：10
Credits：0.5
Course Category：Lecture
Aims & Requirements:
I will systematically explain a new approach to large deviation theory for metric space valued Markov processes.This approach is based upon the variational nature of large deviation, and is expressed using the language of Hamilton-Jacobi equation in metric spaces.
The general theory is largely motivated, and will be illustrated, using physical applications, particularly nonlinear PDE models with a statistical mechanics connection.
No prior knowledge of large deviation theory or nonlinear PDE is assumed. I will explain
relevant parts of existing theories and develop new tools as necessary during the instruction.
A first course on measure theory and a first course on functional analysis will be sufficient
for going through the material rigorously.
Primary Coverage：
Part one: Motivation
1. Sanov Theorem from the view of Boltzmann -- an elementary derivation
a. Sanov theorem and Sterling formula
b. Why did Boltzmann care and the concept of entropy
c. Gibbs conditioning, maximum entropy principle and equilibrium stat. mech.
2. Free energy and entropy, through the view of Gibbs
a. An entropy-free energy duality
b. Properties of entropy
Part two: General theory
3. Large deviation for metric space valued random variables
a. Laplace Lemma
b. Large deviation principle and Laplace principle, equivalence and subtleties
4. Large deviation for metric space valued random variables - II
c. Exponential tightness
d. Rate function, and techniques on identifying it
5. Large deviation for stochastic processes
a. Exponential tightness -- a random version of Ascoli-Arzela theorem
b. Rate function/action functional in the context of processes
6. The case of Markov processes
a. Historical examples -- Freidlin-Wentzell; Donsker-Varadhan; and multi-scale examples
b. Martingale problems and an introduction to semigroup techniques
7. The case of Markov processes -II
c. A class of nonlinear semigroups (Nisio semigroup) useful in large deviation theory
d. Hamilton-Jacobi equations, viscosity solutions and maximum principles in abstract settings.
8. The case of Markov processes - III
e. Hamiltonian and Lagrangian operators as time-infinitesimal generator for free energy and entropy
f. Some a priori estimates for resolvents of Hamiltonians
9. The case of Markov processes - IV (compact state space)
g. Convergence of Hamilton-Jacobi equation, the Barles - Perthame procedure in compact state space
h. Convergence of the Nisio semigroup and large deviation for time-marginal distributions
i. A large deviation theorem for Markov processes - A Hamiltonian formalism
9. The case of Markov processes - V (non-locally compact state space)
j. Another priori resolvent estimate, and exponential tightness
k. Barles-Perthame procedure in non-locally compact state spaces
l. Extended large deviation theorem for Markov processes in non-locally compact state spaces.
10. The case of Markov processes - VI (incorporating multi-scale and projections)
m. Examples of averaging, and homogenization, and projection of Markov processes
n. Multi-valued viscosity solution
o.Things we need to modify in the proofs
p. Yet another extension of large deviation theorem for Markov processes.
11. The case of Markov processes - VII
q. Theory of optimal control, and rate function/action functional representation as time integrals
r. Dynamical programming principle, variational form of the Hamiltonian, and the role of Lagrangian
s. Yet yet another extension of the large deviation theorem
12. First order Hamilton-Jacobi equations
t. The important roles of comparison theorem for Hamilton-Jacobi equations in large deviation,
u. Comparison principle for first order Hamilton-Jacobi equations in Euclidean space,
13. First order Hamilton-Jacobi equations - II
v. Examples in Hilbert space and in space of probability measures
w. A comparison theory in metric spaces
Part three: Examples,
14. Small noise SDE
a. Freidlin - Wentzell theory (diffusion with jumps)
b. Averaging and homogenization of diffusions
15. Large time ergodic behavior
a. Donsker - Varadhan theory
b. Conditions and verification of conditions
16. Stochastic PDEs
a. Semi-linear equations (Allen-Cahn and Cahn-Hilliard)
b. Weakly interacting particle systems and comparison principle
Part four: Beyond large deviation, or back to large deviation in the sense of Boltzmann(see Part one)
17. Hamilton-Jacobi equation in space of measures
a. An introduction to modern theory of optimal mass transportation
b. Comparison principle for a Hamilton-Jacobi equation in space of measures
18. Statistical characterization for large time behaviors of a complex flow model
a. 2-D vorticity formulation of Navier-Stokes and a stochastic particle model
b. Finite time large deviation for the model
c. A controlled, mixed Hamiltonian-Gradient flow in space of measures.
19. Statistical characterization for large time behaviors of a complex flow model - II
c. Infinite time limit of the rate function and a stationary Hamilton-Jacobi equation in space of measures
d. Large time limit of entropy and the important role of fluctuation.
e. A conjecture from Onsager, how far are we from it?
20. A compressible, irrotational Euler equation in finite time
a. Variational derivation using mass transport theory
b. Gibbs conditioning for weakly interacting particles
c. The associated Hamilton-Jacobi PDE in space of measures is well posed.
21. Open problems and open directions.
Textbook:
Large deviation for stochastic processes,
Survey and Monograph Series 131, American Mathematical Society, 2006,
by Jin Feng and Thomas G. Kurtz
References:
Dembo and Zeitouni
Large Deviation Techniques and Applications, 2nd Edition, Springer
Freidlin and Wentzell
Random Perturbations of Dynamical Systems
Author：Jing Feng
