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

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.