The KPZ Equation, Non-Equilibrium Energy Solutions, and Weak Universality for Long-Range Interactions

Kevin Yang

Published 2018-10-05Version 1

We study the weak KPZ universality problem by extending the method of energy solutions developed by Goncalves-Jara-Sethuraman \cite{GJS15} and Gubinelli-Perkowski \cite{GP} to interacting particle systems with singular initial data with respect to equilibrium generalizing the fundamental flat initial data. Moreover, we develop these methods to prove the weak KPZ universality conjecture for interacting particle systems with long-range interactions again with respect to the same singular initial data, vastly improving the result for range at most 3 obtained in \cite{DT}. The proof of these results depends on constructing Brownian approximations for the initial data, constructing a coupling for the density fluctuation field, and performing stochastic analysis for the resulting sequence of Cole-Hopf solutions. As a byproduct of our analysis, the method of energy solutions is also adapted to prove weak KPZ universality for the same interacting particle systems but with flat-to-Brownian initial data and its similar generalizations.