Temporal fractal nature of linearized Kuramoto–Sivashinsky SPDEs and their gradient in one-to-three dimensions

Abstract

Let \(U=\{U(t,x), (t,x)\in \mathring{{\mathbb {R}}}_+\times {\mathbb {R}}^d\}\) and \(\partial _{x}U=\{\partial _{x}U(t,x), (t,x)\in \mathring{{\mathbb {R}}}_+\times {\mathbb {R}}\}\) 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 to prove the exact global temporal moduli and temporal LILs for the L-KS SPDEs and gradient, and utilize them to prove that the sets of temporal fast points where exceptional oscillation of \(U(\cdot ,x)\) and \(\partial _{x}U(\cdot ,x)\) occur infinitely often are random fractals, and evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of \(U(\cdot ,x)\) and \(\partial _{x}U(\cdot ,x)\), in time, are everywhere dense with power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension \(\dim _{_{p}}(B)\) of the target set B.