MiS Preprint Repository

We consider the non-linear equation $u+{\partial }_{t}u-{\partial }_x^2\pi (u)=\xi$ driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that ${\pi }^{\prime }$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of ${\pi }^{\prime }$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $\alpha$ for any $\alpha <\frac{1}{2}$ w.\ r.\ t.\ the parabolic metric. More precisely, we show that the corresponding local Hölder norm has almost Gaussian 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 appeal to finite and infinitesimal versions of the $H^{-1}$-contraction principle and a Campanato iteration.