Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

By P. Gonçalves, Milton Jara. 2013

189 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/16e8c147adb1499b95735b0ce739aa7ccf4666a6

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Citations in the corpus

The Dynamic $${\Phi^4_3}$$Φ34 Model Comes Down from Infinity (2016, methodology): “Related results were obtained for the KPZ equation, first in [2] via a Cole–Hopf transformation, and then, following [22], in a series of works including [11,17,19,20,27,29].”
Energy solutions of KPZ are unique (2015, methodology): “time and space derivative respectively. The term ξis a gaussian centered space–time white noise with covariance E[ξ(t,x)ξ(s,y)] = δ(t−s)δ(x−y). Energy solutions were introduced by Gonc¸alves and Jara [3] in order to describe the ﬂuctuations of a large class of weakly asymmetric interacting particle ”
Lectures on singular stochastic PDEs (2015, methodology): “Next we set up the analytical tools we need in the rest of the lectures: Besov spaces and some basic harmonic analysis based on 1The paper [11] is the revised published version of [10].”