# Synchronization for KPZ

## TL;DR

This paper investigates the long-term behavior of KPZ-like equations on a torus driven by ergodic noise, establishing synchronization and a one-force, one-solution principle using infinite-dimensional extensions of random matrix results.

## Contribution

It introduces a novel analysis of KPZ equations' long-time behavior, proving synchronization and solution uniqueness in a new infinite-dimensional framework.

## Key findings

- Established almost sure synchronization with exponential speed
- Proved a one-force, one-solution principle for KPZ equations
- Extended random matrix techniques to infinite-dimensional stochastic PDEs

## Abstract

We study the longtime behavior of KPZ-like equations: $$ \partial_{t}h(t,x) = \Delta_{x} h (t, x) + | \nabla_{x}h (t,x)|^{2} + \eta(t, x), \qquad h(0, x) = h_0(x), \qquad (t, x) \in (0, \infty) \times \mathbb{T}^{d} $$ 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 H\"older spaces.