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
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
Hamilton-Jacobi equation in abstract spaces where the dynamic is
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
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
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.
Recent journal publications:
- On a class
of first order Hamilton-Jacobi
in metric spaces (with Luigi
Ambrosio), Preprint, (2013).
a system of conservation
laws (with Truyen
of Mathematiques Pures et Appliquees, page 318-390, Vol 97
(2012) A short announcement in Comptes
control for a mixed flow of Hamiltonian and gradient type in space of
probability measures (with
press, Trans. A.M.S.
- Small time
asumptotics for fast mean-reverting stochastic volatility models (with Jean-Pierre Fouque and Rohini
Applied Probability, Vol 22, No.4, 1541-1575
- A singular
1-D Hamilton-Jacobi equation, with applications to large deviation of
diffusions (with Xiaoxue
Deng and Yong Liu), Commun. Math. Sci. Vol 1, Issue 9, page
(with Martin Forde and Jean-Pierre Fouque), SIAM
Journal on Financial Mathematics, Vol 1, pp 126-141 (2010).
gradient flows in infinite dimensions
(with Markos Katsoulakis), Archive for Rational Mechanic
and Analysis, vol 192, page 275-310, (2009)
Nualart), Journal of
Functional Analysis, vol 255., page 313-373, (2008).
- Large Deviation and PDE,
Dec., 2011 --- slides
for lecture one.
- Optimal Controlled PDE and
Hamilton-Jacobi equation in Space of Probability Measures,
Institute of Applied Math., Chinese Academy of Sciences, Dec., 2011 and
Hiroshima University, Jan. 2012 --- lecture one, two, three,