  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.
