We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expansion of the densities. In particular, our proofs are based on sharp estimates of the correlation structure of the solution, which may have their own interest.

中文翻译：

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加性白噪声热方程解的二次变化的渐近展开

我们考虑时空白噪声随机热方程解的空间二次变化序列。该序列满足中心极限定理。通过使用 Malliavin 演算，我们通过证明密度序列的收敛性和通过在密度的渐近展开中找到二阶项来改进这个结果。特别是，我们的证明是基于对解决方案的相关结构的精确估计，这可能有他们自己的兴趣。