Hölder regularity for a non-linear parabolic equation driven by space-time white noise

Abstract

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2π(u)=ξ$ driven by space-time white noise $ξ$, which is uniformly parabolic because we assume that $π'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $π'$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $α$ for any $α<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder 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 Hölder norm. In a second step this estimate is improved to the optimal Hölder exponent at the expense of weakening the integrability to stretched exponential.