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Nonlinear fluctuations of weakly asymmetric interacting particle systems. (English) Zbl 1293.35336
Summary: 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.

##### MSC:
 35Q82 PDEs in connection with statistical mechanics 35Q53 KdV equations (Korteweg-de Vries equations) 82C22 Interacting particle systems in time-dependent statistical mechanics 35B65 Smoothness and regularity of solutions to PDEs
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