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L-Kuramoto-Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp Hölder regularity, and Swift-Hohenberg law equivalence. (English) Zbl 1321.35267

Summary: 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 \(\{\mathbb{R}_+ \times \mathbb{R}^d \}_{d = 1}^3\). 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 \(\varepsilon_1\) to the L-KS spatial operator and \(\varepsilon_2\) to the noise term, we show that the dimension-dependent critical ratio \(\varepsilon_2 / \varepsilon_1^{d / 8}\) controls the limiting behavior of the L-KS SPDE, as \(\varepsilon_1, \varepsilon_2 \searrow 0\); 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 Hölder regularity – of the Swift-Hohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R11 Fractional partial differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
45H05 Integral equations with miscellaneous special kernels
45R05 Random integral equations
60J60 Diffusion processes
60J65 Brownian motion
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