Milton Jara

Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion

Abstract We prove a nonequilibrium central limit theorem for the position of a tagged particle in the one-dimensional nearest neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile ρ 0 : R → [ 0 , 1 ] .

Nonequilibrium Scaling Limit for a Tagged Particle in the Simple Exclusion Process with Long Jumps

We prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium. The scaling limit is a time-inhomogeneous process of independent increments, related to the solution of a fractional heat equation.

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann–Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann–Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.

A stochastic Burgers equation from a class of microscopic interactions

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on Z, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order O(n−γ) for 1/2<γ≤1, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein–Uhlenbeck process. However, at the critical weak asymmetry when γ=1/2, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp “Boltzmann–Gibbs” estimate which improves on earlier bounds.

Quenched scaling limits of trap models

Fix a strictly positive measure

Superdiffusion of Energy in a Chain of Harmonic Oscillators with Noise

W

3/4-fractional superdiffusion in a system of harmonic oscillators perturbed by a conservative noise

on the

Symmetric exclusion as a random environment: hydrodynamic limits

d

A martingale problem for an absorbed diffusion: the nucleation phase of condensing zero range processes

-dimensional torus

Limit theorems for some continuous-time random walks

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