Publications

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 spaceswhich have potential to solve concrete problems.
  Part one (on Large deviation) of June 2015 Marc Kac seminar on Probability and Physics in Utrecht, the Netherlands. (PDF)
  Part two (on Hamilton-Jacobi equations) of the Kac seminar. (PDF)
• 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. 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.