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.

设\(U=\{U(t,x), (t,x)\in \mathring{{\mathbb {R}}}_+\times {\mathbb {R}}^d\}\)且\(\partial _{x}U=\{\partial _{x}U(t,x), (t,x)\in \mathring{{\mathbb {R}}}_+\times {\mathbb {R}}\}\)分别是由一维到三维空间中的时空白噪声驱动的四阶线性化 Kuramoto-Sivashinsky (L-KS) SPDE 的解和梯度解。我们使用底层显式核来证明 L-KS SPDE 和梯度的精确全局时间模量和时间 LIL，并利用它们来证明时间快点集在\(U(\cdot ,x)的异常振荡处\)和\(\partial _{x}U(\cdot ,x)\)无限经常出现是随机分形，并评估它们的豪斯多夫维数和命中概率。已经证实，\(U(\cdot ,x)\)和\(\partial _{x}U(\cdot ,x)\)的这些点在时间上几乎处处稠密，具有连续统的幂当然，它们的命中概率由目标集B的包装维度\(\dim _{_{p}}(B)\)决定。