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

Abstract

We consider the non-linear equation $T^{-1}u+{\partial }_{t}u-{\partial }_x^2\pi (u)=\xi$ 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 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.

arXiv comment54 pages