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Averaging Gaussian functionals

This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati's Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel $γ_0$ is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits Gaussian fluctuation; with some extra mild integrability condition on $γ_0$, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPD