线性随机分数阶扩散方程在时间零点的辛钦与钟氏重对数律
Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation
中文总结 AI 辅助
该研究针对带零初值的线性随机分数阶扩散方程，建立了时间零点的辛钦型重对数律，在特定附加条件下证明了钟氏型重对数律，将随机热方程的初始时间重对数律推广至一类时间分数阶随机扩散方程。
中文摘要 AI 辅助
我们考虑带有零初值的线性随机分数阶扩散方程：∂^β u(t,x)=−(−Δ)^(α/2)u(t,x)+I_t^γ[Ẇ(t,x)]，其中t>0，x∈ℝ^d，α>0，β∈(0,2)，γ≥0。驱动噪声Ẇ是时间上分数阶、空间上具有里斯型协方差的中心高斯广义场。对每个固定的x∈ℝ^d，我们建立时间过程t↦u(t,x)在时间零点的辛钦型重对数律。在附加条件0≤γ<1和β+γ<2+H下，我们还证明了对应的钟氏型重对数律。证明依赖于可调和表示、精确频率截断估计、精确小球渐近以及局部化论证。这些结果将随机热方程的初始时间重对数律推广到了一大类时间分数阶随机扩散方程。
英文摘要
We consider the linear stochastic fractional diffusion equation \begin{equation*} \partial^β u(t,x)=-\left(-Δ\right)^{α/2}u(t,x) +I_t^γ\bigl[\dot W(t,x)\bigr], \qquad t>0,\quad x\in\mathbb R^d, \end{equation*} with zero initial conditions, where $α>0$, $β\in(0,2)$, and $γ\ge0$. The driving noise $\dot W$ is a centered Gaussian generalized field that is fractional in time and has Riesz-type spatial covariance. For each fixed $x\in\mathbb R^d$, we establish a Khinchin-type law of the iterated logarithm at time zero for the temporal process $t\mapsto u(t,x)$. Under the additional conditions $0\leγ<1$ and $β+γ<2+H$, we also prove the corresponding Chung-type law. The proofs rely on a harmonizable representation, sharp frequency-truncation estimates, an exact small-ball asymptotic, and a localization argument. These results extend the initial-time laws of the iterated logarithm for stochastic heat equations to a broad class of time-fractional stochastic diffusion equations.
