  We introduce what we call the second-order Boltzmann-Gibbs principle, which
allows 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.
