Almost Sure Uniform Convergence Of Random Hermite Series

We continue the analysis of random series associated to the multidimensional harmonic oscillator $-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$ on ${\mathbb{R}}^d$ with d \geq 2$$\mathrm{.}Morepreciselyweobtainanecessaryandsufficientconditiontogetthealmostsureuniformconvergenceonthewholespace$$\mathbb{R}$^d$\mathrm{.}Itturnsoutthatthesameconditiongivesthealmostsureuniformconvergenceonthesphere$$\mathbb{S}$^{d-1}$(despite$$\mathbb{S}$^{d-1}$isazeroLebesguemeasureof$$\mathbb{R}$^d$)\mathrm{.}Fromaprobabilisticpointofview,ourproofadaptsastrategyusedbythefirstauthorforboundarylessRiemanniancompactmanifolds\mathrm{.}However,ourproofrequiressharpoff-diagonalestimatesofthespectralfunctionof$-Δ+ |x|^2$ . Such estimates are obtained using elementary tools.