# A stochastic Burgers equation from a class of microscopic interactions

@article{Gonccalves2012ASB, title={A stochastic Burgers equation from a class of microscopic interactions}, author={Patr'icia Gonccalves and Milton Jara and Sunder Sethuraman}, journal={arXiv: Probability}, year={2012} }

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 $1/2<\gamma\leq1$, 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…

## 62 Citations

On Microscopic Derivation of a Fractional Stochastic Burgers Equation

- Mathematics, Physics
- 2014

We derive, from a class of asymmetric mass-conservative interacting particle systems on $${\mathbb{Z}}$$Z, with long-range jump rates of order | · |−(1+α) for 0 < α < 2, different continuum…

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

- Mathematics, Physics
- 2017

We consider the weakly asymmetric simple exclusion process on the discrete space $\{1,...,n-1\}$, in contact with stochastic reservoirs, both with density $\rho\in{(0,1)}$ at the extremity points,…

Convergence to the Stochastic Burgers Equation from a degenerate microscopic dynamics

- Mathematics
- 2016

In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we…

Stochastic Burgers equation from long range exclusion interactions

- Mathematics
- 2016

We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form $p_n(\cdot)=s(\cdot)+\gamma_na(\cdot)$, such that its symmetric part $s(\cdot)$ is…

On energy solutions to stochastic Burgers equation.

- Mathematics
- 2020

In this review we discuss the weak KPZ universality conjecture for a class of 1-d systems whose dynamics conserves one or more quantities. As a prototype example for the former case, we will focus on…

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

- Mathematics
- 2014

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…

Nonlinear fluctuations of interacting particle systems

- Physics
- 2013

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, onedimensional stochastic process by a possibly nonlinear function…

The infinitesimal generator of the stochastic Burgers equation

- MathematicsProbability Theory and Related Fields
- 2020

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal…

The KPZ Equation, Non-Stationary Solutions, and Weak Universality for Finite-Range Interactions

- Mathematics
- 2018

We study the weak KPZ universality problem by extending the KPZ universality results for weakly asymmetric exclusion processes to non-simple variants under deterministic initial data with constant…

Stochastic PDE Limit of the Six Vertex Model

- Mathematics, Physics
- 2018

We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $$\Delta \rightarrow 1^+$$ Δ → 1 + so as to zoom into the ferroelectric/disordered phase…

## References

SHOWING 1-10 OF 68 REFERENCES

Stochastic Burgers and KPZ Equations from Particle Systems

- Mathematics
- 1997

Abstract: We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that,…

Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries

- Mathematics
- 2006

We consider attractive particle systems in $${\mathbb {Z}^d}$$ with product invariant measures. We prove that when particles are restricted to a subset of $${\mathbb {Z}^d}$$ , with birth and death…

Convergence towards Burger's equation and propagation of chaos for weakly asymmetric exclusion processes

- Mathematics
- 1987

We consider a nearest neighbor exclusion process on with generator G[var epsilon] = Gs + [var epsilon]Ga, where Gs and Ga denote the generators of a symmetric and a totally asymmetric exclusion…

The Kardar-Parisi-Zhang Equation and Universality Class

- 2011

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or…

A Central Limit Theorem for the Weakly Asymmetric Simple Exclusion Process

- Mathematics
- 1991

We consider the one-dimensional weakly asymmetric nearest neighbour exclusion process and study, in macroscopic space-time coordinates, the fluctuations of the associated density field around the…

Hydrodynamic limit for a particle system with degenerate rates

- Mathematics
- 2007

We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong…

The stochastic heat equation: Feynman-Kac formula and intermittence

- Mathematics
- 1995

We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a…

Macdonald processes

- Mathematics, Physics
- 2014

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $$q,t \in…

The Crossover Regime for the Weakly Asymmetric Simple Exclusion Process

- Mathematics, Physics
- 2010

We consider the asymmetric simple exclusion process in one dimension with weak asymmetry (WASEP) and 0–1 step initial condition. Our interest are the fluctuations of the time-integrated particle…

A central limit theorem for reversible exclusion and zero-range particle systems

- Mathematics
- 1996

We give easily verifiable conditions under which a functional central limit theorem holds for additive functionals of symmetric simple exclusion and symmetric zero-range processes. Also a reversible…