Current and density fluctuations for interacting particle systems with anomalous diffusive behavior

On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions

The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive(homogenisation)limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a b…

谐势中的广义 Langevin 方程：反常扩散与能量均分

The generalized Langevin equation in harmonic potentials: Anomalous diffusion and equipartition of energy

本文研究了谐势中具幂律记忆核的广义 Langevin 方程，发现粒子位移可呈扩散或超扩散，并证明在完全单调函数类核下系统满足能量均分条件，推广了已有结果。

关于一个动力学弱相互作用粒子系统的扩散-平均场极限

On the diffusive-mean field limit of a kinetic weakly interacting particle system

研究弱相互作用动力学朗之万动力学系统的扩散-平均场极限，将前人结果扩展到亚椭圆/亚强制情形，通过格林-库博等公式计算极限布朗运动协方差矩阵，分析\(O(2)\)模型，证明低温且有相变时两极限可能不可交换。

Diffusive fluctuations for one-dimensional totally asymmetric interacting random dynamics

We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under…

Equilibrium fluctuations for diffusive symmetric exclusion with long jumps and infinitely extended reservoirs

We provide a complete description of the equilibrium fluctuations for diffusive symmetric exclusion processes with long jumps in contact with infinitely extended reservoirs and prove that they behave as generalized Ornstein-Uhlenbeck processes with various boundary conditions, depending mainly on the strength of the reservoirs. On the way, we also give a general statement about uniqueness of the Ornstein-Uhlenbeck process originated by the microscopic dynamics of the underlying interacting particle systems and adapt it to our study.

Heat kernel upper bounds for interacting particle systems

We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

Conservative interacting particles system with anomalous rate of ergodicity

We analyze certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence.