  The asymmetric simple exclusion process and its analysis by mode coupling
theory (MCT) is reviewed. To treat the weakly asymmetric case at large space
scale $x\varepsilon^{-1}$, %(corresponding to small Fourier momentum at scale
$p\varepsilon$), large time scale $t \varepsilon^{-\chi}$ and weak hopping bias
$b \varepsilon^{\kappa}$ in the limit $\varepsilon \to 0$ we develop a
mesoscale MCT that allows for studying the crossover at $\kappa=1/2$ and
$\chi=2$ from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality.
The dynamical structure function is shown to satisfy for all $\kappa$ an
integral equation that is independent of the microscopic model parameters and
has a solution that yields a scale-invariant function with the KPZ dynamical
exponent $z=3/2$ at scale $\chi=3/2+\kappa$ for $0\leq\kappa<1/2$ and for
$\chi=2$ the exact Gaussian EW solution with $z=2$ for $\kappa>1/2$. At the
crossover point it is a function of both scaling variables which converges at
macroscopic scale to the conventional MCT approximation of KPZ universality for
$\kappa<1/2$. This fluctuation pattern confirms long-standing conjectures for
$\kappa \leq 1/2$ and is in agreement with mathematically rigorous results for
$\kappa>1/2$ despite the numerous uncontrolled approximations on which MCT is
based.
