Research

Interests:

My current interests focus on problems relating to continuum mechanics, probability, scaling and variational nonlinear PDEs. In the grand scheme of things, I am interested in mathematical questions with a physical origin.

At a technical level, I have been working on Hamilton-Jacobi theories of various kinds that describes limit (particle number as well as time) behavior of infinite particles. Although the ultimate goal is to understand how (and if) probabilistic behaviors arise from deterministic systems, at present, the pictures are not complete yet. My current investigation focuses on using deterministic averaging theories in producing probabilistic links in effective, macroscopic models. My main techniques are taken from Probability (stochastic calculus and limit theorems) and Analysis (PDEs, optimal control, mass transport theory, metric space analysis).

Three lines of significant works in past research:

  1. Developed a Hamilton-Jacobi method for large deviation of metric space valued Markov process [Feng-Kurtz 2006].
  2. Introduced an idea of formulating Hamilton-Jacobi in the space of probability measures, and developed well-posedness theories for a class of such equations [Chapters 9.1, 9.4 and 13.3 of Feng-Kurtz 2006, Feng-Katsoulakis 2009 which was written earlier, and Ambrosio-Feng 2014 for models directly applicable to variational formulation in continuum mechanics].
  3. Borrowing ideas from entropy/renormalized solution from nonlinear PDE theory, discovered a stochastic version of it for a class of stochastic scalar conservation laws with the source terms being random. Such solution has a satisfactory well-posedness theory [Feng-Nualart 2008].

Selected Publications:

  • A book: Large Deviations for Stochastic Processes (with T.G. Kurtz), Mathematical Surveys and Monographs Vol. No. 131, American Mathematical Society, (2006) 410 pages.
  • A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion Arxiv: 1903.00052  (with T. Mikami and J. Zimmer), Communications in Mathematical Physics. Vol. 385, (2021), 1-54. https://doi.org/10.1007/s00220-021-04110-1
  • On a class of first order Hamilton-Jacobi equations in metric spaces (with L. Ambrosio), J. Diff. Equations. Vol. 256, No. 7. (2014), 2194-2245.
  • A comparison principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions (with M. Katsoulakis), Archive for Rational Mechanics and Analysis, Vol. 192 (2009), 275-310.
  • Stochastic scalar conservation laws (with D. Nualart), Journal of Functional Analysis, Vol. 55, No. 2 (2008), 313-373.
  • Large deviation for diffusions and Hamilton-Jacobi equation in Hilbert spaces, The Annals of Probability, Vol. 34, No.1 (2006), 321-385.
  • Large deviation for stochastic Cahn-Hilliard equation, Methods of Functional Analysis and Topology, Vol. 9, No. 4 (2003), 333-356.

Current focus:

  • Hydrodynamic limit derivations of asymptotically non-interacting deterministic particles through Hamilton-Jacobi theory.  An outline of the program is announced in a four-page publication in Oberwolfach Reports Vol. 17, Issue 2/3, 2020. DOI: 10.4171/OWR/2020/29
  • A change of coordinate methods for Hamilton-Jacobi equations in geodesic metric spaces;