My primary research areas are applied probability and applied dynamical systems. I use various rigorous and computational methods to study "large" dynamical systems arising from physics and biology. These systems are usually high dimensional, noisy, and nonequilibrium.
I: Nonequilibrium Statistical Mechanics
The derivation of macroscopic thermodynamic laws from microscopic Hamiltonian dynamics is a long standing challenge to mathematicians. Nonequilibrium thermodynamic models are also representative examples of "large" dynamical systems. I am interested in a variety of problems arising from nonequilibrium statistical mechanics, such as nonequilibrium kinetic particle systems, stochastic stability of nonequilibrium steady-states (NESS), and the microscopic derivation of macroscopic thermodynamic laws.
- Thermal conductivity and local thermodynamic equilibrium of stochastic energy exchange models (with Wenbo Xie), Under Review (pdf)
- From billiards to thermodynamic laws: stochastic energy exchange model (with Lingchen Bu), Chaos: An Interdisci- plinary Journal of Nonlinear Science, accepted, (pdf)
- On the polynomial convergence rate to nonequilibrium steady-states, Annals of Applied Probability, accepted (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)
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II: Mathematical neuroscience
Understanding how cortex processes informations is an incredible complex task. From a mathematical point of view, neuronal networks are another representative examples of "large" dynamical systems that are both interesting and mathematically tractable. I am interested in nonlinear interaction of neurons in spiking neuronal networks, information-theoretic features of complex networks, and large scale simulations related to the visual cortex.
- Firing rate and spatial correlation in a stochastic neural field model (with Hui Xu), Under Review , (pdf)
- How well do reduced models capture the dynamics in models of interacting neurons? (with Logan Chariker and Lai-Sang Young), Journal of Mathematical Biology, accepted (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)
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III, Analysis and computation of invariant probability measures
The study of invariant probability measures of stochastic processes plays an important row in understanding noisy dynamics. Numerical computation is necessary to tackle various complex problems. I am interested in providing analytical and computational tools for the study of invariant probability measures of stochastic differential equations. Our research projects include concentration estimates of invariant measures and advanced numerical methods of solving Fokker-Planck equations.
- A data-driven method for the steady state of randomly purterbed dynamics, Under Review , (pdf)
- Numerical Simulation of Polynomial-Speed Convergence Phenomenon (with H. Xu), Journal of Statistical Physics, 169(4), 2017 (pdf)
- 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)
- 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), Journal of Differential Equations 261, 2552-2583 (pdf)