Jin Feng

Phone 785-864-3764
Email  jfeng AT math.ku.edu

In a nutshell, I have been interested in questions such as the following:  what is the large time most likely structure for a given stochastic model? or slightly more generally, given an observation (mostly a rare event), what is the new large time most likely structure given that such a rare event has occured?  At a more challenging level, the question now becomes, given a deterministic dynamic which is defined using a variational problem, with probability now replaced by cost of actions, what are the analogous questions to the above?

A particular example that fits the above description is a rigorous verification of the Onsager-Joyce-Montgomery theory regarding large time coherent structures of 2-D vortex flows. Progress has been slow but it is being made rigorously now, in joint works with collaborators. Beyond this example, a unifying principle also seems to exist.   In the context of stochastic models, the variational problem usually comes from a large deviation scaling limit.  In (deterministic) continuum mechanics context,  this can also come from a direct variational formulation of the model. A key to the program is to understand behaviors of Hamilton-Jacobi equation in abstract spaces where the dynamic is defined. One example of such space is the Wasserstein space of probability measures, which itself exhibits many peculiar properties.

In earlier work with Tom Kurtz, we developed some known connections between Hamilton-Jacobi equation and the probabilistic large deviation theory for diffusions/jump process to a full program for large deviation of Markov processes in general metric spaces. A key assumption is the associated Hamilton-Jacobi equation being well-posed. Henceforth, I have been working on Hamilton-Jacobi equations and optimal controls in Wasserstein space of probability measures, and in general metric spaces. New theories regarding the meaning and well posedness for a large class of viscosity solution have been developed. Recent and future works will focus on large time behaviors of Lagrangian dynamics in Wasserstein space and in geodesic metric spaces.

Some of the techniques involved are Markov processes theory, limit theory in probability, viscosity solution for Hamilton-Jacobi equations and optimal control, stochastic differential equations and PDEs,  singular perturbations and averaging, conservation laws, theory of optimal mass transportation, large time behavior of Hamilton-Jacobi equations. There are also interesting connections with financial mathematics.

 
 


Teaching 

A book:

  Recent journal publications: Some notes: