AI 中文总结
研究一维时空分数阶随机热方程温和解的空间平均，利用Malliavin-Stein方法结合相关精确估计，证明固定\(t>0\)时归一化空间平均的定量中心极限定理及泛函中心极限定理。
AI 中文摘要
我们研究由时空白噪声驱动的一维时空分数阶随机热方程温和解的空间平均。对于固定的\(t>0\)，我们证明了在\([-R,R]\)上归一化空间平均的定量中心极限定理：当\(R\to\infty\)时，其分布在全变差距离下以\(R^{-1/2}\)的速率收敛到标准正态分布。证明依赖于Malliavin-Stein方法，并结合了时空分数阶热核和温和解的Malliavin导数的精确估计。我们还建立了一个泛函中心极限定理。
英文摘要
We study spatial averages of the mild solution to a one-dimensional space--time fractional stochastic heat equation driven by space--time white noise. For fixed \(t>0\), we prove a quantitative central limit theorem for the normalized spatial average over \([-R,R]\): as \(R\to\infty\), its law converges to the standard normal law at rate \(R^{-1/2}\) in total variation distance. The proof relies on the Malliavin--Stein method, combined with precise estimates for the space--time fractional heat kernel and for the Malliavin derivative of the mild solution. We further establish a functional central limit theorem.
