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

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We consider the non-linear equation T⁻¹ u+∂ₜu-∂ₓ²π(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<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.