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

TL;DR

This paper establishes exact moduli of continuity and regularity properties for L-Kuramoto-Sivashinsky SPDEs and time-fractional SPIDEs driven by space-time white noise, revealing criticality and differences in temporal laws.

Contribution

It provides the first precise moduli of continuity and regularity results for these classes of SPDEs, highlighting the critical nature of the half-derivative SPIDE and differences in temporal behavior.

Findings

Both classes are $(1/2)^-$ H"older continuously differentiable in space when $d=1$

Exact uniform and local moduli of continuity for the gradient are obtained

The half-derivative SPIDE is a critical case with rougher spatial regularity

Abstract

We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (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 Brownian-time-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 H\"older regularity for the above two classes of equations in $d=1,2,3$. We show that both classes are $(1/2)^-$ H\"older 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 H\"older 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 Houdr\'e-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 versions of (1) and (2) on compacts via change of measure.