L-Kuramoto–Sivashinsky SPDEs vs. time-fractional SPIDEs: Exact continuity and gradient moduli, 1/2-derivative criticality, and laws

Abstract

Abstract. We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-KuramotoSivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by space-time white noise in one-to-three dimensional space. Both classes were introduced-with Browniantime-type kernel formulations-by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above two classes of equations in d = 1, 2, 3. We show that both classes are (1/2) − Hölder continuously differentiable in space when d = 1, and we give the exact uniform and local moduli of continuity for the gradient in both cases. This is unprecedented for SPDEs driven by space-time white noise. Our results on exact moduli show that the half-derivative SPIDE is a critical case. It signals the onset of rougher modulus regularity in space than both time-fractional SPIDEs with time-derivatives of order < 1/2 and L-KS SPDEs. This is despite the fact that they all have identical spatial Hölder regularity, as shown earlier by Allouba. Moreover, we show that the temporal laws governing (1) and (2) are fundamentally different. We relate L-KS SPDEs to the Houdré-Villa bifractional Brownian motion, yielding a Chung-type law of the iterated logarithm for these SPDEs. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on delicate sample path properties of Gaussian random fields. On the other hand, it builds on and complements Allouba's earlier works on (1) and (2). Similar regularity results hold for Allen-Cahn nonlinear members of (1) and (2) on compacts via change of measure.2010 Mathematics Subject Classification. 60H15, 60H20, 60H30, 45H05, 45R05, 35R60, 60J45, 60J35, 60J60, 60J65.

Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

“…As in [4], for the LKS-SPDE, we use the LKS kernel to define their rigorous mild SIE formulation. This LKS kernel, as shown in as in [1][2][3], is the fundamental solution to the deterministic version of (12) (a ≡ 0 and b ≡ 0) below, and is given by:…”

“…Notation 1. Positive and finite constants (independent of x) in Section i are numbered as c i,1 , c i,2 , .... We conclude this section by citing the following spatial Fourier transform of the (ε, ϑ) LKS kernels from Lemma 2.1 in [4].…”

“…The fourth order linearized Kuramoto-Sivashinsky (LKS) SPDEs are related to the model of pattern formation phenomena accompanying the appearance of turbulence (see [1][2][3][4] for the LKS class and for its connection to many classical and new examples of deterministic and stochastic pattern formation PDEs, and see [5,6] for classical examples of deterministic and stochastic pattern formation PDEs).…”

“…The existence/uniqueness as well as sharp dimension-dependent L p and Hölder regularity of the linear and nonlinear noise version of (1) were investigated in [1,2,9,10]. It was studied in [4] that exact uniform and local moduli of continuity for the LKS-SPDE in the time variable t and space variable x, separately. In fact, it was established in [4] that exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for the fourth order the LKS-SPDEs and their gradient.…”

“…It was studied in [4] that exact uniform and local moduli of continuity for the LKS-SPDE in the time variable t and space variable x, separately. In fact, it was established in [4] that exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for the fourth order the LKS-SPDEs and their gradient. It was studied in [11] that the solution to a stochastic heat equation with the space-time white noise in time has infinite quadratic variation and is not a semimartingale, and also investigated temporal central limit theorems for modifications of the quadratic variation of the stochastic heat equation with space-time white noise in time.…”