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

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 fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the Swift-Hohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal H\"older regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on $\{\Rp\times\Rd\}_{d=1}^{3}$. The spatio-temporal H\"older exponents are exactly the same as the striking ones we proved for our recently introduced Brownian-time Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schr\"odinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters $\vepo$ to the L-KS spatial operator and $\vept$ to the 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 less regular second order heat SPDEs. Finally, we give a change-of-measure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift-Hohenberg and the law equivalence---and hence the same H\"older regularity---of the Swift-Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.