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Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work,we give a novel explicit-kernel formulation useful for a large class of fourthorder deterministic, stochastic, linear, and nonlinear PDEs in multispatialdimensions. These include pattern formation equations like the Swift-Hohenbergand many other prominent and new PDEs. We first establish existence,uniqueness, and sharp dimension-dependent spatio-temporal H\"older regularityfor the canonical (zero drift) L-KS SPDE, driven by white noise on$\{\Rp\times\Rd\}_{d=1}^{3}$. The spatio-temporal H\"older exponents areexactly the same as the striking ones we proved for our recently introducedBrownian-time Brownian motion (BTBM) stochastic integral equation, associatedwith time-fractional PDEs. The challenge here is that, unlike the positive BTBMdensity, the L-KS kernel is the Gaussian average of a modified, highlyoscillatory, and complex Schr\"odinger propagator. We use a combination ofharmonic and delicate analysis to get the necessary estimates. Second,attaching order parameters $\vepo$ to the L-KS spatial operator and $\vept$ tothe noise term, we show that the dimension-dependent critical ratio$\vept/\vepo^{d/8}$ controls the limiting behavior of the L-KS SPDE, as$\vepo,\vept\searrow0$; and we compare this behavior to that of the lessregular second order heat SPDEs. Finally, we give a change-of-measureequivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. Inparticular, we prove uniqueness in law for the Swift-Hohenberg and the lawequivalence---and hence the same H\"older regularity---of the Swift-HohenbergSPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.

L-Kuramoto-Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp H\"older regularity, and Swift-Hohenberg law equivalence