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 $\{\text{Rp}\times \text{Rd}{\}}_{d=1}^3$. 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 $\text{vepo}$ to the L-KS spatial operator and $\text{vept}$ to the noise term, we show that the dimension-dependent critical ratio $\text{vept}\mathrm{/}{\text{vepo}}^{d\mathrm{/}8}$ controls the limiting behavior of the L-KS SPDE, as $\text{vepo},\text{vept}\searrow 0$; 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.