Averaging Gaussian functionals

Abstract

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 γ0 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 γ0, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.

Chaos expansion of uniformly distributed random variables and application to number theory

“…Let U be given by (10). Then for every k > 0 the random variable U 2k admits the following Wiener chaos expansion:…”

“…, we obtain the chaos decomposition of the random variable cos(T log pU ) where T > 0, p is a prime number and U is defined in (10). We actually have…”

“…Let U be given by (10). Assume that for every T > 0 the square integrable random variable Then for every n ≥ 1, K n (T ) → T →∞ 0 in L 2 ( ).…”

“…In many situations, for such a limit theorem, there exists a dominant chaos for X T (see, e.g., Theorem 1.7 in [10] for an example on the average of solutions to some stochastic partial differential equations, or [8] for an example related to stochastic geometry). That is, there exists N 0 ≥ 1 such that I N 0 (f N 0 (T )) converges in law, as T → ∞, to N(0, 1) while the other chaos components are negligible, i.e.…”

Spatial averages for the parabolic Anderson model driven by rough noise

“…These techniques are not available in our framework because the noise is colored in time, which forces to restrict our study to linear equations, where we can use Wiener chaos expansions and Feynman-Kac formulas. In Nualart and Zheng (2020a), that is the closest work to the present paper, Nualart and Zheng consider the PAM (1.1) on R + × R d with the correlation kernels γ 0 and γ satisfying the following conditions:…”