本論文研究在隨機場影響下，熱方程於時間與空間同時進行尺度縮放時的漸近行為。我們關注縮放後解的機率分布極限以及其誤差收斂性，並透過兩個主要定理加以分析。首先，在一維情況下，我們證明當時間與空間依特定比例縮放時，熱方程解的機率分布將收斂至常態分布，且誤差隨縮放趨近於零。其次，我們探討在固定時間下，不同空間位置上解的聯合分布，並分別採用兩種方法分析其極限分布，證明兩者均具有相同的誤差收斂速率。本研究有助於理解具長程相關性之隨機初始條件下，隨機偏微分方程解的波動行為與高斯近似特性。

This thesis investigates the asymptotic behavior of the heat equation under the influence of a random field when both time and space undergo parabolic scaling. We analyze the limiting distribution of the rescaled solution and the convergence of the associated approximation error through two main theorems. First, in the one-dimensional case, we prove that the probability distribution of the solution converges to a normal distribution under a specific scaling of time and space, with the approximation error vanishing in the limit. Second, we study the joint distribution of the solution at two distinct spatial locations at a fixed time. Two methods are employed to analyze the limiting behavior, and both yield the same error convergence rate. This research contributes to understanding fluctuation behavior and Gaussian approximation in stochastic partial differential equations with long-range dependent random initial data.

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