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Gaussian weights on fractals and Edelman's ghosts and shadows

Primarily AI-generated textHuman understanding: no partsmath.PR — Probabilitymath.MG — Metric Geometry

Contributed by Djalil Chafaï ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

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 β>1\beta>1, we construct a closed cone of Hausdorff dimension β\beta whose Gaussian-weighted β\beta-dimensional Hausdorff measure has exactly the χβ\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 β≥1\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.

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References

  1. F. Baudoin and C. Lacaux, Fractional Gaussian fields on the Sierpiński gasket and related fractals , J. Anal. Math. 146 (2022), 719--739. doi:10.1007/s11854-022-0211-x (https://doi.org/10.1007/s11854-022-0211-x). arXiv:2003.04408 (https://arxiv.org/abs/2003.04408).arXivDOI
  2. I. Dumitriu and A. Edelman, Matrix models for beta ensembles , J. Math. Phys. 43 (2002), no. 11, 5830--5847. doi:10.1063/1.1507823 (https://doi.org/10.1063/1.1507823). arXiv:math-ph/0206043 (https://arxiv.org/abs/math-ph/0206043).arXivDOI
  3. A. Edelman, The random matrix technique of ghosts and shadows , conference note, version dated November 22, 2009, 6 pp. https://math.mit.edu/~edelman/publications/random_matrix_technique.pdf.link
  4. J. F. C. Kingman, Random walks with spherical symmetry , Acta Math. 109 (1963), 11--53. doi:10.1007/BF02391808 (https://doi.org/10.1007/BF02391808).DOI
  5. W. Matysiak, Hypergroups and quantum Bessel processes of non-integer dimensions , arXiv:1611.04819 (https://arxiv.org/abs/1611.04819), 2016 (version with corrections to the published article). doi:10.1007/s10959-016-0688-2 (https://doi.org/10.1007/s10959-016-0688-2).arXivDOI
  6. J. Pitman and M. Winkel, Squared Bessel processes of positive and negative dimension embedded in Brownian local times , Electron. Commun. Probab. 23 (2018), no. 74, 1--13. doi:10.1214/18-ECP174 (https://doi.org/10.1214/18-ECP174). arXiv:1804.07316 (https://arxiv.org/abs/1804.07316).arXivDOI
  7. M. Rösler, Dunkl operators: theory and applications , in Orthogonal Polynomials and Special Functions (E. Koelink and W. Van Assche, eds.), Lecture Notes in Math. 1817 , Springer, 2003, 93--135. doi:10.1007/3-540-44945-0_3 (https://doi.org/10.1007/3-540-44945-0_3). arXiv:math/0210366 (https://arxiv.org/abs/math/0210366).arXivDOI
  8. M. Rösler, Bessel convolutions on matrix cones , Compos. Math. 143 (2007), no. 3, 749--779. doi:10.1112/S0010437X06002594 (https://doi.org/10.1112/S0010437X06002594).DOI

Version history

  1. v1Submitted by Djalil ChafaïInitial depositCurrentOct 18, 2026