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Almost Sure Uniform Convergence Of Random Hermite Series (2506.03858v1)

Published 4 Jun 2025 in math.FA

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.

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