Hamilton-Jacobi from large deviation theory, statistical mechanics and classical continuum mechanics

Wednesday, August 16, 2017 - 10:00am - 10:45am
Lind 409
Jin Feng (University of Kansas)
In 2006, Tom Kurtz and I co-authored a book on a Hamilton-Jacobi approach to the Large deviation theory for Markov processes. An important step is to prove the uniqueness of a limiting Hamilton-Jacobi PDE. I will outline this approach in the beginning part of this talk. Then I give a few examples, including one open problem in statistical mechanics. A characteristic difficulty of the open problem is an invariance relation.

In the rest of the talk, I turn to a simpler example in classical continuum mechanics where infinitely many deterministic particles weakly interact through a mean field. This problem also has a hidden invariance. I will describe how a well-posedness theory was eventually achieved for the associated Hamilton-Jacobi equation (in the space of probability measures). It is a bit surprising that a classical naive definition of Hamiltonian needs to be augmented to give the well-posedness.