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

Abstract

We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. 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 the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient.

Asymptotic Distributions for Power Variations of the Solution to the Spatially Colored Stochastic Heat Equation

“…en, (9) has been extended by Corcuera et al [15], Nourdin [17], Dobrushin and Major [18], Taqqu [19], Breuer and Major [20], Giraitis and Surgailis [21], Wang [22], and Wang and Wang [23]. Swanson [13] extended (9) to modifications of the quadratic variation of the solutions of SHE driven by space-time white noise.…”

Temporal Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

“…The authors of [6] investigated the exact, dimension-dependent, spatiotemporal, and uniform and local moduli of continuity for the L-KS SPDE in the time variable t and space variable x. The authors of [11] investigated the solutions to Equation (1) in time and possessing an infinite quadratic variation. Temporal asymptotic distributions for the realized power variations of the L-KS SPDEs Equation (1) were investigated in [11].…”

“…The authors of [11] investigated the solutions to Equation (1) in time and possessing an infinite quadratic variation. Temporal asymptotic distributions for the realized power variations of the L-KS SPDEs Equation (1) were investigated in [11]. These results naturally cause the following motivating questions:…”