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

Wensheng Wang,Changkai Zhou

doi:10.3390/sym13071306

Abstract

Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for (t,x)∈R+×R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for U(·,x) and ∂xU(·,x). It has been confirmed that almost all sample paths of U(·,x) and ∂xU(·,x), in time, are nowhere differentiable.

Highlights

Let U = U(t, x) for (t, x) ∈ R+ × Rd and ∂xU = ∂xU(t, x) for (t, x) ∈ R+ × R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively
We are concerned with delicate regularity properties of paths of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces
Equation (8) describes the size of the minimum oscillation of the gradient of L-KS SPDE solution ∂xU(·, x) over the compact rectangle Itime is (h)

Summary

Introduction

We are concerned with delicate regularity properties of paths of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces. We provide exact, dimension-dependent, and temporal moduli of non-differentiability for the important class of stochastic equations:. In [6], the following exact temporal Chung’s LIL for L-KS SPDE U(t, x) and the gradient process ∂xU(t, x) are obtained. This article is devoted to establishing the following exact temporal moduli of non-differentiability for L-KS SPDE U(t, x) and the gradient process ∂xU(t, x). Equation (8) describes the size of the minimum oscillation of the gradient of L-KS SPDE solution ∂xU(·, x) over the compact rectangle Itime is (h) (up to a constant factor). It quantifies precisely the roughness of the sample paths of U(·, x) by β(h) For this reason, the function β(h) is referred to as a modulus of non-differentiability of the L-KS SPDE solution. The mild formulation of (1) is obtained by setting σ ≡ 1 and a ≡ 0 in (11)

Temporal Spectral Density for L-KS SPDEs and Their Gradient

Bifractional Brownian Motion Link for L-KS SPDEs and Their Gradient

Extremes for L-KS SPDEs and Their Gradient

Conclusions