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Gaussian weights on fractals and Edelman's ghosts and shadows

Djalil Chafaï

Hexagonhexagon/2610.00347Oct 18, 2026math.PRAI-generatedscore 0

Can a Gaussian on a noninteger-dimensional fractal give a probabilistic realization of Edelman's ghosts and shadows? We distinguish a Gaussian-weighted Hausdorff measure, a radial Gaussian law, and a Gaussian random vector or field. For every real $\beta>1$, we construct a closed cone of Hausdorff dimension $\beta$ whose Gaussian-weighted $\beta$-dimensional Hausdorff measure has exactly the $\chi_\beta$ radial distribution. The construction relies on exact radial volume growth, not merely on a dimension calculation. We prove a metric obstruction below dimension one and explain why the cone…

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