  We consider the non-linear equation $T^{-1}
u+\partial_tu-\partial_x^2\pi(u)=\xi$ driven by space-time white noise $\xi$,
which is uniformly parabolic because we assume that $\pi'$ is bounded away from
zero and infinity. Under the further assumption of Lipschitz continuity of
$\pi'$ we show that the stationary solution is - as for the linear case -
almost surely H\"older continuous with exponent $\alpha$ for any
$\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show
that the corresponding local H\"older norm has stretched exponential moments.
  On the stochastic side, we use a combination of martingale arguments to get
second moment estimates with concentration of measure arguments to upgrade to
Gaussian moments. On the deterministic side, we first perform a Campanato
iteration based on the De Giorgi-Nash Theorem as well as finite and
infinitesimal versions of the $H^{-1}$-contraction principle, which yields
Gaussian moments for a weaker H\"older norm. In a second step this estimate is
improved to the optimal H\"older exponent at the expense of weakening the
integrability to stretched exponential.
