Synchronization for KPZ

Abstract

We study the longtime behavior of KPZ-like equations: $${\partial }_{t}h(t,x)={\Delta }_{x}h(t,x)+|{\nabla }_{x}h(t,x){|}^2+\eta (t,x),h(0,x)=h_0(x),(t,x)\in (0,\infty )\times {\mathbb{T}}^d$$ on the $d-$dimensional torus ${\mathbb{T}}^d$ driven by an ergodic noise $\eta$ (e.g. space-time white in $d=1$. The analysis builds on infinite-dimensional extensions of similar results for positive random matrices. We establish a one force, one solution principle and derive almost sure synchronization with exponential deterministic speed in appropriate Hölder spaces.

arXiv comment35 Pages