Asymptotic expansion for the quadratic variations of the solution to the heat equation with additive white noise

Abstract

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.