H\"older regularity for a non-linear parabolic equation driven by space-time white noise

TL;DR

This paper proves that solutions to a non-linear parabolic SPDE driven by space-time white noise are almost surely Hölder continuous with exponent less than 1/2, using advanced probabilistic and PDE techniques.

Contribution

It establishes Hölder regularity for non-linear stochastic PDEs with space-time white noise, extending known results from linear cases and employing novel combination of probabilistic and PDE methods.

Findings

Stationary solutions are almost surely Hölder continuous with exponent < 1/2.

Local Hölder norm has stretched exponential moments.

Gaussian moments obtained for a weaker Hölder norm.

Abstract

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.