We consider the one-dimensional stochastic heat equation driven by a multiplicative space-time white noise. We show that the spatial integral of the solution from $-R$ to $R$ converges in total variance distance to a standard normal distribution as $R$ tends to infinity, after renormalization. We also show a functional version of this central limit theorem.

中文翻译：

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随机热方程的中心极限定理

我们考虑由乘法时空白噪声驱动的一维随机热方程。我们表明，从 $-R$ 到 $R$ 的解决方案的空间积分在总方差距离上收敛于标准正态分布，因为在重整化后 $R$ 趋于无穷大。我们还展示了这个中心极限定理的函数版本。