My primary research areas are applied dynamical systems and applied probability. I work with random perturbed dynamical systems, and stochastic processes with a dynamical systems flavor, using both analytical and computational methods.
Nonequilibrium Statistical Mechanics
I am interested in a variety of problems arising from nonequilibrium statistical mechanics, such as the existence and uniqueness of nonequilibrium steady-states (NESS), exponential/polynomial convergence towards steady-states, and the microscopic derivation of thermodynamic laws. I am also interested in "nonequilibrium" neural field models.
- How well do reduced models capture the dynamics in models of interacting neurons? (with Logan Chariker and Lai-Sang Young), Under review (pdf)
- On the polynomial convergence rate to nonequilibrium steady-states, Under review (pdf)
- Polynomial Convergence to Equilibrium for a System of Interacing Particles (with Lai-Sang Young), Annals of Applied Probability, 27(1), 2017, 65-90 (pdf)
- Local Thermodynamic Equilibrium for some Multidimensional Stochastic Models (with Peter Nandori and Lai-Sang Young) , Journal of Statistical Physics, 163(1), 61-91 (pdf)
- On the stochastic behaviors of locally confined particle systems, Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 073121(2015) (pdf)
- Nonequilibrium steady states for a class of particle systems (with Lai-Sang Young), Nonlinearity 27, page 607, 2014 (pdf)
- Existence of nonequilibrium steady state for a simple model of heat conduction (with Lai-Sang Young), Journal of Statistical Physics, pages 1 -- 24, 2013 (pdf)
Stochastic Differential Equations and Systematic Measures of Complex Networks
I am interested in invariant probability measures of stochastic differential equations. We obtained some new estimates of invariant probability measures of stochastic differential equations and applied them to complex biological networks. A class of systematic measures of biological network proposed in the literature of system biology, including degeneracy, complexity and robustness, are rigorously investigated and quantified.
- Systematic measures of biological networks, part I: Invariant measures and entropy (with Yingfei Yi), Communications on Pure and Applied Mathematics, LXIX, 1777-1811. 2016 (pdf)
- Systematic measures of biological networks, part II: Degeneracy, complexity and robustness. (with Yingfei Yi), Communications on Pure and Applied Mathematics, LXIX, 1952-1983, 2016 (pdf)
- Quantification of degeneracy in bio- logical systems for characterization of functional interactions between modules (with G. Dwivedi, W. Huang, M. Kemp and Y. Yi), Journal of theoretical biology, 302:2938, 2012 (pdf)
- Numerical Simulation of Polynomial-Speed Convergence Phenomenon (with H. Xu), Journal of Statistical Physics, 169(4), 2017 (pdf)
- A fast exact simulation algorithm for a class of Markov jump processes (with L. Hu), Journal of Chemical Physics, 143(18), 2015 (pdf)
- A limiting strategy for the back and forth error compensation and correction method for solving advection equations (with L. Hu, Y. Liu), Mathematics of Computation 85 (2016), 1263-1280 (pdf)
Fokker-Planck Equations on Discrete Spaces
- Fokker-Planck equations for a free energy functional or Markov process on a graph (with S-N. Chow, W. Huang and H-M. Zhou), Archive for Rational Mechanics and Analysis 203.3 (2012): 969-1008. (pdf)
- A free energy based mathematical study for molecular motors (with S-N. Chow, W. Huang and H-M. Zhou), Regular and Chaotic Dynamics 16.1-2 (2011): 117-127. (pdf)
- Convergence to global equilibrium for Fokker-Planck equations on a graph and talagrand-type inequalities (with R. Che, W. Huang and P. Tetali), Under Review (pdf)