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math.PR — Probability

Gaussian weights on fractals and Edelman's ghosts and shadows

Contributed by Djalil Chafaï

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 does not supply ordinary Gaussian addition or a scalar algebra. A separate Gram-determinant argument rules out preserving all polynomially continued Euclidean Gaussian Gram identities in a positive model. We then give proofs of the angular beta-gamma identities, associativity of Bessel-Kingman convolution, and its Gaussian convolution semigroup for $\beta\geq1$. Squared Bessel processes, tridiagonal and bidiagonal matrix models, Dunkl Gaussians, and Gaussian fields on fractals provide complementary constructions with explicitly different meanings of dimension. None of these statements asserts the existence of a full noninteger-dimensional scalar algebra.