A stochastic Burgers equation from a class of microscopic interactions

Abstract

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^{-\gamma })$ for $1/2<\gamma \le 1$, 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 $\gamma =1/2$, 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.

Published in at http://dx.doi.org/10.1214/13-AOP878 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Crossover to the stochastic Burgers equation for the WASEP with a slow bond
• Non-equilibrium Fluctuations of Interacting Particle Systems
• ASEP(q,j) converges to the KPZ equation
• Open ASEP in the Weakly Asymmetric Regime
• On microscopic derivation of a fractional stochastic Burgers equation
• Some recent progress in singular stochastic PDEs
• KPZ fluctuations in finite volume
• Universality of deterministic KPZ
• Local KPZ behavior under arbitrary scaling limits
• Hairer-Quastel universality in non-stationarity via energy solution theory
• Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes
• Second order Boltzmann-Gibbs principle for polynomial functions and applications
• Comments on David Aldous and Persi Diaconis' "Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem"
• Non-Stationary KPZ equation from ASEP with slow bonds
• Probabilistic approach to the stochastic Burgers equation
• On energy solutions to stochastic Burgers equation
• Stochastic Burgers Equation via Energy Solutions from Non-Stationary Particle Systems
• Quadratic Fluctuations of the Simple Exclusion Process
• Large time behavior of strong solutions for stochastic Burgers equation
• A Microscopic Derivation Of Coupled Spde's With A Kpz Flavor
• Characterization of Gradient Condition for Asymmetric Partial Exclusion Processes and Their Scaling Limits