Spatial integral of the solution to SPDEs with time-independent noise

Abstract: In this talk, we review some recent results regarding the asymptotic behavior of the spatial integral of the solution to the hyperbolic/parabolic Anderson model, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a spatially homogeneous Gaussian noise. The noise does not depend on time, which means that Itô's martingale theory for stochastic integration cannot be used. Using the methodology initiated in Huang, Nualart and Viitasaari (2020), which consists of a combination of Malliavin calculus techniques with Stein's method for normal approximations, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance.

This talk is based on joint work with Wangjun Yuan.

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