Almost Sure Uniform Convergence Of Random Hermite Series

TL;DR

This paper establishes a precise criterion for the almost sure uniform convergence of random Hermite series on both the entire space and spheres, extending previous methods with new spectral estimates.

Contribution

It provides the first necessary and sufficient condition for uniform convergence of these series on unbounded domains, adapting techniques from Riemannian geometry.

Findings

Condition for almost sure uniform convergence on  and spheres

Spectral function estimates using elementary tools

Extension of probabilistic convergence results to  harmonic oscillator

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.