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

Abstract

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 Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on $\{\Rp\times\Rd\}_{d=1}^{3}$. The spatio-temporal Hölder 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 Schrödinger 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 Hölder regularity---of the Swift-Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.

31 page, 2 Appendices. v6 identical to v5 (only fixed title appearance (no \scriptsize)). v5 is the final version incorporating referee's suggestions: cosmetic changes in title & abstract, expanded discussion of second result (Theorem 1.2 Sec 1.4.2) & included a comparison to the heat SPDE, the Swift-Hohenberg result is now separate (Corollary 1.1), & other minor enhancements & typos corrections

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