Almost Sure Uniform Convergence Of Random Hermite Series

Abstract

We continue the analysis of random series associated to the multidimensional harmonic oscillator $-\Delta +|x{|}^2$ on ${\mathbb{R}}^d$ with d \geq 2$$. More precisely we obtain a necessary and sufficient condition to get the almost sure uniform convergence on the whole space ${\mathbb{R}}^d$ . It turns out that the same condition gives the almost sure uniform convergence on the sphere ${\mathbb{S}}^{d-1}$ (despite ${\mathbb{S}}^{d-1}$ is a zero Lebesgue measure of ${\mathbb{R}}^d$). From a probabilistic point of view, our proof adapts a strategy used by the first author for boundaryless Riemannian compact manifolds. However, our proof requires sharp off-diagonal estimates of the spectral function of $-\Delta +|x{|}^2$ . Such estimates are obtained using elementary tools.