### Interests:

My current interests run along two lines: 1. rational continuum mechanics, 2. stochastic analysis.

More specifically, goal to the first line is to rigorously derive continuum level equations of hydrodynamics (and thermodynamics next) through first principles. I take an approach that relies upon mathematical tools such as Hamilton-Jacobi theory in calculus of variations, optimal control and PDEs; averaging theory for Hamiltonian systems; Stochastics; Mass transport theory and first order calculus in metric spaces. Regarding the second line, I am interested in a number of singular stochastic PDEs with only first order derivatives (hence no regularization effects in the usual sense) or with totally non-linearity in the equation (hence regularization effect, if any, is expressed in more subtle ways). These are usually equations modeling physical systems under random influences. In recent years, I entered a very critical stage of development for the first topic, hence have not been able to spend enough time with the second one.

A common theme in all these works is that I explore the maximum principle in either direct or hidden abstract ways.

Three lines of significant works in past research:

- Developed a Hamilton-Jacobi method for large deviation of metric space valued Markov process [Feng-Kurtz 2006].
- 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 explained more in detail in Feng-Nguyen 2012 (J. of Math. Pures Appl.), and Ambrosio-Feng 2014 for models directly applicable to variational formulation in continuum mechanics].
- Found a successful implementation of renormalized solution idea from deterministic nonlinear PDE theory to stochastic settings, illustrated through stochastic scalar conservation laws. Such approach 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.
*Large deviations from a nonlinear semigroup convergence and variational point of view. Starting from martingale-problem formulation of Markov processes and viscosity solution for Hamilton-Jacobi equations, this book develops both stochastic as well as PDE methods**in generic metric spaces**which have potential to solve concrete problems.*

Part two (on Hamilton-Jacobi equations) of the Kac seminar. - 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
*Another title of this can be large deviation for diffusive scaling limit of the stochastic Carleman models. Section 4 of this work introduces a new method for the averaging step of stochastic hydrodynamic derivations. The new method is most naturally viewed from perspective of Hamiltonian dynamical systems through the weak KAM theory.* - 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.
*For certain class of equations, it is possible to define viscosity solution and get well-posedness theory for Hamilton-Jacobi PDE in generic geodesic metric spaces (free of any curvature assumption). The results are applied in Section 4 to understand equations arising from action minimization formulation of compressible flows (in this case the space is non-negatively curved in Alexandrov sense).* - 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.
*This paper showed that we can develop a uniqueness viscosity theory for Hamilton-Jacobi equations in “very bad” spaces such as the space of probability measures. Such space does not have Radon-Nikodyn property as previous literature pioneered by Crandall-Lions required. The new method relies upon modern mass transport theory. Illustrations using three examples gradually move from good (Hilbert spaces) to the very bad spaces ( probability measures). The examples show how a physically motivated structure regarding fluctuations and entropy-entropy dissipation relation induced some inequalities, making the above claims became theorems.* - Stochastic scalar conservation laws (with D. Nualart), Journal of Functional Analysis, Vol. 55, No. 2 (2008), 313-373.
*Shows that Kruzkov’s entropy solution (and later renormalized solution) ideas have a complete analogy in stochastic situations once the stochastic equations are properly re-formulated. A companion stochastic compensated-compactness method is also developed.* - Large deviation for diffusions and Hamilton-Jacobi equation in Hilbert spaces, The Annals of Probability, Vol. 34, No.1 (2006), 321-385.
*The paper treats models of stochastic evolution equations with examples including stochastic Allen-Cahn equation. An earlier method by Tataru-Crandall-Lions is adapted to ensure uniqueness of the corresponding Hamilton-Jacobi equations in Hilbert spaces.* - Large deviation for stochastic Cahn-Hilliard equation, Methods of Functional Analysis and Topology, Vol. 9, No. 4 (2003), 333-356.
*An illustration of the Hamilton-Jacobi approach to large deviation applied to stochastic Cahn-Hilliard equation, with a detailed treatment on uniqueness theory for Hamilton-Jacobi PDEs with field-valued variables modeled in Hilbert space H-1*.

### 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.