Lu Xu

Gran Sasso Science Institute | GSSI · Mathematics

We study the evolution in equilibrium of the fluctuations for the conserved quantities of a chain of anharmonic oscillators in the hyperbolic space-time scaling limit. Boundary conditions are determined by applying a constant tension at one side, while the position of the other side is kept fixed. The Hamiltonian dynamics is perturbed by random ter...

We study the quasi-static limit for the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document} entropy weak solution of scalar one-dimensional hype...

We study the one-dimensional asymmetric simple exclusion process on the lattice {1,⋯,N}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1, \dots ,N\}$$\end{document} w...

We study the hydrodynamic behaviour of the asymmetric simple exclusion process on the lattice of size n. In the bulk, the exclusion dynamics performs rightward flux. At the boundaries, the dynamics is attached to reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are weakened by \(n^\theta \) for some \(\theta <0\) and (2)...

We consider a scalar conservation law with relaxation in a bounded domain Ω, which describes the macroscopic evolution of some interacting particle system. The strength of the relaxation grows to infinity at ∂Ω. We define the entropy solution u ∈ L ∞ and prove that it is unique. When the relaxation is integrable, u satisfies the boundary condition...

We consider the equilibrium perturbations for two stochastic systems: the $d$-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-\alpha}$ to the equilibrium profile, and speed up the process by $N^{1+\kappa}$ for parameters $0<\kappa\le\alpha$. Under some additional...

We study the hydrodynamic behaviour of the asymmetric simple exclusion process on the lattice of size $n$. In the bulk, the exclusion dynamics performs rightward flux. At the boundaries, the dynamics is attached to reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are weakened by $n^\theta$ for some $\theta<0$ and (2) the...

We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice. The particles can be created/annihilated at the boundaries with time-dependent rate. These boundary dynamics are properly accelerated. We prove the hydrodynamic limit of the particle density profile, under the hyperbolic space-time rescaling, evolves with the...

We study the quasi-static limit for the one-dimensional asymmetric simple exclusion process with open boundaries. The quasi-stationary profile evolves with the quasi-static Burgers equation.

We study the quasi-static limit for the $L^\infty$ entropic weak solution of the one-dimensional Burgers' equation with boundary conditions. The quasistationary profile evolves with the quasi-static Burgers' equation, whose entropic solution is determined by the stationary profile corresponding to the boundary data at a given time.

We consider a chain of n coupled oscillators placed on a one-dimensional lattice with periodic boundary conditions. The interaction between particles is determined by a weakly anharmonic potential V_n = r^2/2 + sigma_nU(r), where U has bounded second derivative and sigma_n vanishes as n goes to infinite. The dynamics is perturbed by noises acting o...

In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our resu...

We study the macroscopic behavior of the fluctuations in equilibrium for the conserved quantities of an anharmonic chain of oscillators under hyperbolic scaling of space and time. Under a stochastic perturbation of the dynamics conservative of such quantities, we prove that these fluctuations evolve macroscopically following the linearized Euler sy...

We investigate the asymptotic behaviors of the solution to a stochastic heat equation with a periodic, gradient-type nonlinear term. We extend the central limit theorem for finite-dimensional diffusions to infinite-dimensional settings. We also verify the tightness and present an invariance principle .