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\numberwithin{equation}{section}
\hypersetup{
  pdftitle={Gaussian weights on fractals and Edelman's ghosts and shadows},
  pdfsubject={Fractal cones, chi distributions, Bessel--Kingman convolution, and beta ensembles}
}

\title[Fractal Gaussians, ghosts, and shadows]{Gaussian weights on fractals and Edelman's ghosts and shadows}
\author{ChatGPT prompted by Djalil Chafaï}
\date{October 1 2026}

\begin{document}

\begin{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 $\beta>1$, 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.
\end{abstract}

\maketitle

\section{The question and the conventions}
\label{sec:scope}

Edelman's note \cite{Edelman2009} introduces isotropic ghosts through radial
laws and calls observable real or complex quantities their shadows. For a
standard Gaussian ghost with dimension parameter $\beta$, the norm shadow has
law $\chi_\beta$. Section~6 of that note explicitly considers fractal theory
in connection with a volume-scaling rule, but does not construct a fractal
model. The note also separates operations on independent ghosts from the
more difficult issue of dependence.

The question considered here is therefore twofold. Can a fractal carry a
natural measure whose Gaussian weighting gives the prescribed norm? Can the
same construction realize additional shadows and their joint algebraic
relations? The first question has an exact affirmative answer for
$\beta>1$. The second requires additional structure and cannot be answered
by a Hausdorff-dimension calculation alone.

Throughout, $\operatorname{Gamma}(a,\theta)$ denotes the gamma distribution
with shape $a>0$ and scale $\theta>0$, having density
\[
  \frac{x^{a-1}e^{-x/\theta}}{\Gamma(a)\theta^a},\qquad x>0.
\]
For $\beta>0$ and $t>0$, let $\gamma_{\beta,t}$ be the probability measure on
$[0,\infty)$ with density
\begin{equation}
\label{eq:chi-density}
  \gamma_{\beta,t}(dr)
  =\frac{r^{\beta-1}e^{-r^2/(2t)}}
  {2^{\beta/2-1}\Gamma(\beta/2)t^{\beta/2}}\,dr,
  \qquad r>0.
\end{equation}
Thus $R\sim\gamma_{\beta,t}$ means
\begin{equation}
\label{eq:gamma-square}
  R\overset{d}{=}\sqrt{t}\,\chi_\beta,
  \qquad R^2\sim\operatorname{Gamma}(\beta/2,2t).
\end{equation}
We set $\gamma_{\beta,0}=\delta_0$. Direct integration gives
\begin{equation}
\label{eq:radial-moments}
  \mathbb E R^p
  =(2t)^{p/2}\frac{\Gamma((\beta+p)/2)}{\Gamma(\beta/2)},
  \qquad p>-\beta.
\end{equation}
In particular, $\mathbb E R^2=\beta t$. At integer $\beta=d$, this is the
norm distribution of an $N(0,tI_d)$ vector.

We use $\mathcal H^s$ for Hausdorff measure, with any one fixed standard
normalization. Its normalization affects some constants below, but not the
probability measures obtained after normalization. All restrictions of
Hausdorff measure below are to Borel sets. No fractional-dimensional
Lebesgue measure or exterior algebra is assumed.

\section{The exact radial-volume criterion}
\label{sec:radial}

\subsection{Gaussian weighting of a metric measure space}
\label{prop:radial}
\noindent\textbf{Proposition.}
Let $(F,d)$ be a metric space with a distinguished point $o$, and let $\mu$
be a Borel measure. Write $\rho(x)=d(o,x)$. Suppose that, for constants
$C\in(0,\infty)$ and $\beta>0$,
\begin{equation}
\label{eq:volume}
  \mu\{x:\rho(x)\leq r\}=Cr^\beta,
  \qquad r>0.
\end{equation}
Then, for every $t>0$,
\begin{equation}
\label{eq:weighted-gaussian}
  \nu_t(dx)=\frac{1}{Z_t}e^{-\rho(x)^2/(2t)}\,\mu(dx),
  \qquad
  Z_t=C\beta\,2^{\beta/2-1}\Gamma(\beta/2)t^{\beta/2},
\end{equation}
is a probability measure, and
\begin{equation}
\label{eq:radial-shadow}
  \rho_{\#}\nu_t=\gamma_{\beta,t}.
\end{equation}
Conversely, if the Gaussian weighting of a Borel measure is defined with
$0<Z_t<\infty$ and satisfies \eqref{eq:radial-shadow} for one $t>0$, then
\eqref{eq:volume} holds for some $C\in(0,\infty)$.

\begin{proof}
The pushforward $\eta=\rho_{\#}\mu$ has distribution function
$\eta([0,r])=Cr^\beta$. Continuity from above at zero gives
$\eta(\{0\})=0$, and subtraction on intervals gives
\[
  \eta(dr)=C\beta r^{\beta-1}\,dr.
\]
Consequently,
\[
  Z_t=C\beta\int_0^\infty r^{\beta-1}e^{-r^2/(2t)}\,dr
  =C\beta\,2^{\beta/2-1}\Gamma(\beta/2)t^{\beta/2}.
\]
The same pushforward calculation gives \eqref{eq:radial-shadow}.
Conversely,
\[
  \rho_{\#}\nu_t(dr)=Z_t^{-1}e^{-r^2/(2t)}\eta(dr).
\]
Since the weight is strictly positive, \eqref{eq:chi-density} implies
\[
  \eta(dr)=
  \frac{Z_t}{2^{\beta/2-1}\Gamma(\beta/2)t^{\beta/2}}
  r^{\beta-1}\,dr.
\]
Integrating from zero to $r$ proves the converse.
\end{proof}

\subsection{What a dimension calculation does not imply}

For a general radial pushforward $\eta=\rho_{\#}\mu$, the radius under
Gaussian weighting has measure
\begin{equation}
\label{eq:general-radius}
  Z_t^{-1}e^{-r^2/(2t)}\eta(dr).
\end{equation}
If $V(r)=\mu\{\rho\leq r\}$, then $\eta$ is the Lebesgue--Stieltjes
measure associated with $V$, including any mass at zero. Neither
$\dim_{\mathrm H}F=\beta$ nor two-sided bounds
$c_1r^\beta\leq V(r)\leq c_2r^\beta$ force the exact law
\eqref{eq:chi-density}. Proposition~\ref{prop:radial} identifies the exact
necessary and sufficient radial condition for this weighting procedure.

The weighted half-line
\begin{equation}
\label{eq:half-line}
  \bigl([0,\infty),|r-s|,r^{\beta-1}\,dr\bigr)
\end{equation}
already satisfies that condition for every $\beta>0$. Its support, with
the displayed metric, has Hausdorff dimension one, regardless of $\beta$.
Thus a volume-growth exponent about a distinguished point need not be the
Hausdorff dimension of the support.

\section{A construction using Hausdorff measure on a fractal cone}
\label{sec:cones}

\subsection{Regular angular sets}
\label{lem:angular-sets}
\noindent\textbf{Lemma.}
For every $s>0$, there is a compact set $K$ in a Euclidean unit sphere
and a finite Borel measure $\sigma_0$ with support $K$ such that
\begin{equation}
\label{eq:ahlfors}
  c\varepsilon^s\leq
  \sigma_0\bigl(K\cap B(\theta,\varepsilon)\bigr)
  \leq C\varepsilon^s,
  \qquad \theta\in K,\quad 0<\varepsilon\leq\operatorname{diam}K,
\end{equation}
for some $0<c\leq C<\infty$. In particular,
$\dim_{\mathrm H}K=s$ and $0<\mathcal H^s(K)<\infty$.

\begin{proof}
First suppose $0<s<1$. Set $a=2^{-1/s}<1/2$, and form the compact Cantor
set in $[0,1]$ generated by $x\mapsto ax$ and $x\mapsto1-a+ax$.
Give each interval at level $j$ mass $2^{-j}=a^{js}$. Choose $j$ so that
$a^j\leq\varepsilon<a^{j-1}$. The level-$j$ interval containing a given
point is contained in the ball of radius $\varepsilon$ about that point.
Also, such a ball meets only a bounded number of level-$j$ intervals,
with a bound depending on $a$: distinct level-$j$ intervals are separated
by at least $(1-2a)a^{j-1}$. These observations give \eqref{eq:ahlfors}.

For noninteger $s=k+u$, where $k\geq0$ is an integer and $0<u<1$, take the
product of $[0,1]^k$ with the preceding $u$-dimensional Cantor set,
equipped with the product measure. For integer $s\geq1$, use the cube
$[0,1]^s$. Product balls for the maximum metric give the required bounds;
the Euclidean metric is comparable to that metric. A bounded subset of
$\mathbb R^\ell$ embeds bi-Lipschitzly into the unit sphere in
$\mathbb R^{\ell+1}$ by
\[
  u\longmapsto\frac{(u,1)}{\sqrt{1+|u|^2}}.
\]
This preserves the two-sided estimates, up to constants.

For completeness, the measure bounds imply the Hausdorff-measure assertion
as follows. The upper bound on the measure of balls gives a positive lower
bound for the $s$-dimensional cost of any sufficiently fine cover of $K$.
For the reverse estimate, a maximal family of disjoint balls of radius
$\varepsilon/2$, centered in $K$, has at most a constant times
$\varepsilon^{-s}$ members by the lower measure bound. The corresponding
balls of radius $\varepsilon$ cover $K$, giving a uniform upper bound on
its $s$-dimensional covering cost. Hence $0<\mathcal H^s(K)<\infty$,
which also gives its Hausdorff dimension.
\end{proof}

\subsection{The cone theorem}
\label{thm:cone}
\noindent\textbf{Theorem.}
Fix $\beta>1$, and choose a compact $(\beta-1)$-regular angular set
$K\subset\mathbb S^{m-1}$ as in Lemma~\ref{lem:angular-sets}. Define
\begin{equation}
\label{eq:cone}
  F=\{r\theta:r\geq0,\ \theta\in K\}\subset\mathbb R^m,
  \qquad \mu=\mathcal H^\beta\!\restriction_F.
\end{equation}
Then $F$ is closed, $\dim_{\mathrm H}F=\beta$, and $\mu$ is locally
finite with support $F$. There is a constant $C\in(0,\infty)$ such that
\begin{equation}
\label{eq:cone-volume}
  \mathcal H^\beta(F\cap\overline B(0,r))=Cr^\beta,
  \qquad r>0.
\end{equation}
Consequently the probability measure
\begin{equation}
\label{eq:cone-gaussian}
  \nu_t(dx)=\frac{1}{Z_t}e^{-|x|^2/(2t)}
  \mathcal H^\beta\!\restriction_F(dx)
\end{equation}
satisfies $|X|\sim\gamma_{\beta,t}$ when $X\sim\nu_t$.

\begin{proof}
Closedness follows from compactness of $K$: a convergent sequence of
nonzero cone points has convergent radii, and, if the limiting radius is
positive, its normalized directions converge in $K$.

On every annulus with $0<a\leq r\leq b<\infty$, the polar map is
bi-Lipschitz. Indeed, for unit vectors $\theta,\varphi$,
\begin{equation}
\label{eq:polar-distance}
  |r\theta-u\varphi|^2
  =(r-u)^2+ru|\theta-\varphi|^2.
\end{equation}
Since $a^2\leq ru\leq b^2$, this is comparable to the squared product
metric on $[a,b]\times K$. The product of Lebesgue measure on $[a,b]$
and the regular angular measure has two-sided ball bounds of exponent
$1+(\beta-1)=\beta$. The covering argument in
Lemma~\ref{lem:angular-sets} therefore shows that each nondegenerate
closed annular piece of $F$ has positive finite $\mathcal H^\beta$
measure, and has Hausdorff dimension $\beta$.

Let $A=F\cap\{1/2\leq|x|\leq1\}$. The sets $2^{-j}A$, $j\geq0$,
cover $F\cap\overline B(0,1)$ except for the origin. Hausdorff measure
scales under similarities, so
\[
  0<\mathcal H^\beta(A)
  \leq\mathcal H^\beta(F\cap\overline B(0,1))
  \leq\mathcal H^\beta(A)\sum_{j=0}^\infty2^{-j\beta}<\infty.
\]
The origin has zero $\mathcal H^\beta$ measure. Thus the middle quantity
is an admissible constant $C$.

Since $rF=F$ and $\mathcal H^\beta(rE)=r^\beta\mathcal H^\beta(E)$,
\eqref{eq:cone-volume} follows. This proves local finiteness. The
annular regularity gives positive measure to every relative neighborhood
of every nonzero cone point, and \eqref{eq:cone-volume} gives the same
conclusion at zero. Thus $\mu$ has support $F$.

Finally, $F$ is a countable union of annular pieces and the origin, so
its Hausdorff dimension is exactly $\beta$. Applying
Proposition~\ref{prop:radial} proves the assertion about \eqref{eq:cone-gaussian}.
\end{proof}

For $1<\beta<2$, the angular set can be a regular Cantor set embedded in
an arc of the circle, so the cone lies in the plane. For $\beta=1$, the
ordinary line with its one-dimensional Hausdorff measure gives the usual
one-dimensional Gaussian. Theorem~\ref{thm:cone} concerns positive measures
on actual Euclidean subsets, not a formal continuation of measures.

\subsection{A precise angular disintegration}
\label{prop:polar}

There is a probability measure $\sigma$ on $K$ such that
\begin{equation}
\label{eq:cone-polar}
  \int_F f(x)\,\mu(dx)
  =C\beta\int_K\int_0^\infty
    f(r\theta)r^{\beta-1}\,dr\,\sigma(d\theta)
\end{equation}
for every nonnegative Borel function $f$. In particular, under
\eqref{eq:cone-gaussian}, the variables $|X|$ and $X/|X|$ are independent,
with respective laws $\gamma_{\beta,t}$ and $\sigma$.

To prove this, for a Borel set $E\subset K$ define
\[
  \sigma(E)=C^{-1}\mu\{x:0<|x|\leq1,\ x/|x|\in E\}.
\]
Homogeneity gives
\[
  \mu\{x:0<|x|\leq r,\ x/|x|\in E\}=Cr^\beta\sigma(E).
\]
Subtracting this identity at two radii proves \eqref{eq:cone-polar} on
polar rectangles. Equality of the resulting sigma-finite measures and
then integration prove the general formula.

This argument does not identify $\sigma$ with normalized
$\mathcal H^{\beta-1}\!\restriction_K$, nor is such an identification
needed. More importantly, it does not identify scalar products of
independent $\sigma$-distributed directions with any universal angular
law.

\section{Geometric and algebraic limitations}
\label{sec:limitations}

\subsection{A metric obstruction below dimension one}
\label{prop:metric-obstruction}
\noindent\textbf{Proposition.}
Let $(F,d)$ have Hausdorff dimension strictly smaller than one. For any
$o\in F$, the distance from $o$ of a random point of $F$ cannot have an
absolutely continuous probability distribution on $[0,\infty)$.

\begin{proof}
The map $\rho(x)=d(o,x)$ is $1$-Lipschitz, so
\[
  \dim_{\mathrm H}\rho(F)\leq\dim_{\mathrm H}F<1.
\]
Hence $\rho(F)$ is contained in a Borel subset of the real line having
Lebesgue measure zero. Every distance distribution is concentrated on
that null set and is therefore singular with respect to Lebesgue measure.
\end{proof}

In particular, for $0<\beta<1$ one cannot simultaneously require
$\dim_{\mathrm H}F=\beta$, interpret the ghost norm as distance in that
same metric, and obtain $\chi_\beta$. This does not exclude weighted
spaces such as \eqref{eq:half-line}, arbitrary measurable observables,
or other notions of effective dimension.

\subsection{A Gaussian weight is not Gaussian vector addition}
\label{prop:vector-stability}

For independent cone-valued random variables $X$ and $Y$, ordinary
Euclidean addition gives
\begin{equation}
\label{eq:cone-addition}
  |X+Y|^2=|X|^2+|Y|^2+
  2|X||Y|\left\langle\frac{X}{|X|},\frac{Y}{|Y|}\right\rangle.
\end{equation}
The last factor depends on the angular measure. The sum also need not
belong to the cone. For example, choose $K$ in a cap where
$\theta_1\geq c>0$. For $X,Y$ independent with law $\nu_1$,
\[
  \mathbb E|X+Y|^2=2\beta+2|\mathbb EX|^2>2\beta.
\]
Thus even its second radial moment differs from that of
$\sqrt2\,\chi_\beta$. Symmetrizing a cone removes this particular
nonzero-mean obstruction, but does not determine its angular laws.

There is a stronger obstruction to ordinary vector-valued Gaussian
stability. If an $\mathbb R^m$-valued random variable $X$ has finite
second moment and, for independent copies,
\begin{equation}
\label{eq:vector-stability}
  X_1+X_2\overset{d}{=}\sqrt2\,X,
\end{equation}
then $X$ is a centered, possibly degenerate, Gaussian vector. Indeed,
expectations in \eqref{eq:vector-stability} force $\mathbb EX=0$.
Iteration shows that the normalized sum of $2^j$ independent copies
has the same law as $X$. The multivariate central limit theorem identifies
this law with $N(0,\operatorname{Cov}X)$.

A Gaussian vector has an affine subspace as its support: it is an affine
image of a standard Gaussian in an integer-dimensional space. Therefore
a law with full support equal to a noninteger-dimensional cone cannot
satisfy \eqref{eq:vector-stability}. This is an obstruction to ordinary
vector addition, not to a convolution defined directly on radial laws.

\subsection{A positivity obstruction for continued Gram identities}
\label{prop:gram}
\noindent\textbf{Proposition.}
Fix a noninteger $\beta>0$. There cannot exist, for every integer $k\geq1$,
a random positive semidefinite real $k\times k$ matrix $W_k$ with integrable
determinant such that
\begin{equation}
\label{eq:continued-gram}
  \mathbb E\det W_k=\prod_{j=0}^{k-1}(\beta-j).
\end{equation}
Consequently, no model of real Hilbert-space-valued random vectors can
preserve all the Euclidean Gaussian Gram-determinant identities after
polynomial continuation of the ambient dimension to $\beta$.

\begin{proof}
For an integer $d\geq k$, let $G$ be a $d\times k$ matrix of independent
$N(0,1)$ entries. Cauchy--Binet gives
\[
  \det(G^{\mathsf T}G)
  =\sum_{\substack{I\subset\{1,\ldots,d\}\\|I|=k}}(\det G_I)^2.
\]
For a square $k\times k$ Gaussian matrix, expansion of both determinants
shows $\mathbb E(\det G_I)^2=k!$: distinct permutations have zero mixed
expectation, and each coincident pair contributes one. Hence
\begin{equation}
\label{eq:integer-gram}
  \mathbb E\det(G^{\mathsf T}G)
  =k!\binom dk=\prod_{j=0}^{k-1}(d-j).
\end{equation}
The right side is a polynomial in $d$, so its polynomial continuation
has the value appearing in \eqref{eq:continued-gram}.

Now set $k=\lfloor\beta\rfloor+2$. Exactly one factor in
\eqref{eq:continued-gram} is negative, and none is zero. Its right side
is therefore negative, whereas positive semidefiniteness implies
$\det W_k\geq0$ almost surely. This is a contradiction.
\end{proof}

For example, $\beta=3/2$ and $k=3$ would give
$\mathbb E\det W_3=-3/8$. The hypothesis here is explicitly the
preservation of \emph{all} the identities \eqref{eq:integer-gram}.
It is not a consequence of specifying only $\chi_\beta$ radii or
selected pairwise shadows. Nor is $\beta$ being confused with the
number of vectors $k$. This proposition does not rule out restricted
collections of shadows or the matrix models in Section~\ref{sec:matrices}.

\section{Angular shadows and beta--gamma algebra}
\label{sec:angles}

\subsection{The angular law and its parameter range}
\label{prop:angle}

For $\beta>1$, let $C_\beta$ have density
\begin{equation}
\label{eq:angle-density}
  q_\beta(c)=
  \frac{\Gamma(\beta/2)}{\sqrt\pi\,\Gamma((\beta-1)/2)}
  (1-c^2)^{(\beta-3)/2},\qquad -1<c<1.
\end{equation}
This is normalized by the beta integral, and
\begin{equation}
\label{eq:angle-beta}
  C_\beta^2\sim\operatorname{Beta}\left(\frac12,\frac{\beta-1}{2}\right).
\end{equation}
Its sign is a symmetric random sign independent of its square. If
$(a)_0=1$ and $(a)_j=a(a+1)\cdots(a+j-1)$ for $j\geq1$, then
\begin{equation}
\label{eq:angle-moments}
  \mathbb EC_\beta^{2j}=\frac{(1/2)_j}{(\beta/2)_j},
  \qquad \mathbb EC_\beta^{2j+1}=0.
\end{equation}
For $\beta=1$, define $C_1$ to be a symmetric sign; the same moment
formula holds.

At an integer $d\geq2$, $C_d$ is the first coordinate of a uniform point
on $\mathbb S^{d-1}$, and also the scalar product of two independent
uniform points on that sphere. One way to check the coordinate law is
to normalize $d$ independent standard Gaussians: the square of the first
coordinate is the ratio of independent gamma variables with shapes
$1/2$ and $(d-1)/2$. Orthogonal invariance gives the scalar-product
interpretation. This is the angular variable used in the independent
addition and real-part prescriptions of \cite[Section~3]{Edelman2009}.

The density \eqref{eq:angle-density} is integrable only for $\beta>1$.
Moreover, its proposed moment continuation would give
$\mathbb EC_\beta^2=1/\beta>1$ for $0<\beta<1$, which is impossible
for a random variable in $[-1,1]$. Thus the full parameter range of the
radial law must not be transferred without qualification to this
angular law.

\subsection{A Gaussian coordinate and an independent residual radius}
\label{prop:splitting}
\noindent\textbf{Proposition.}
For $\beta>1$ and $t>0$, let $R\sim\gamma_{\beta,t}$ and let $C_\beta$
be independent of $R$. Then
\begin{equation}
\label{eq:splitting}
  Z=RC_\beta\sim N(0,t),\qquad
  W=R\sqrt{1-C_\beta^2}\sim\gamma_{\beta-1,t},
\end{equation}
and $Z$ and $W$ are independent. At $\beta=1$ the same statement holds
with $W=0$ almost surely.

\begin{proof}
The beta--gamma splitting identity says that, if
$S\sim\operatorname{Gamma}(a+b,\theta)$ and
$U\sim\operatorname{Beta}(a,b)$ are independent, then $SU$ and
$S(1-U)$ are independent gamma variables of respective shapes $a,b$
and common scale $\theta$. Indeed, changing variables from $(S,U)$
to $(v,w)=(SU,S(1-U))$ has inverse Jacobian $1/(v+w)$ and gives the
joint density
\[
  \frac{v^{a-1}e^{-v/\theta}}{\Gamma(a)\theta^a}
  \frac{w^{b-1}e^{-w/\theta}}{\Gamma(b)\theta^b}.
\]
Apply this with $S=R^2$, $U=C_\beta^2$, $a=1/2$,
$b=(\beta-1)/2$, and $\theta=2t$. The independent symmetric sign
of $C_\beta$ then turns $\sqrt{SU}$ into an $N(0,t)$ variable.
The case $\beta=1$ follows by independently signing a half-normal
variable.
\end{proof}

This gives an actual coupling of a norm shadow and a Gaussian real-part
shadow, without a fractal. The residual radius is not an assertion that
there are $\beta-1$ sampled real coordinates. Below dimension one, even
this pair of shadows is impossible under the natural domination condition:
if $R\sim\chi_\beta$, $Z\sim N(0,1)$, and $|Z|\leq R$ almost surely,
then
\begin{equation}
\label{eq:real-part-obstruction}
  1=\mathbb EZ^2\leq\mathbb ER^2=\beta.
\end{equation}
No independence hypothesis is needed for this obstruction.

\section{Bessel--Kingman convolution: an exact radial addition}
\label{sec:hypergroup}

The classical Bessel--Kingman convolution extends addition of independent
isotropic Euclidean vectors at the level of their norm laws
\cite{Kingman1963,Roesler2007}. We give a proof of the properties needed
here. In particular, the extension in the dimension parameter will be
justified by rational identities, not by an unsupported appeal to
analytic continuation from the integers.

\subsection{Definition and associativity}
\label{thm:convolution}
\noindent\textbf{Theorem.}
For $\beta\geq1$ and $r,s\geq0$, set
\begin{equation}
\label{eq:convolution}
  \delta_r *_\beta\delta_s
  =\operatorname{Law}\left(\sqrt{r^2+s^2+2rsC_\beta}\right).
\end{equation}
Extend this operation to Borel probability measures by
\begin{equation}
\label{eq:convolution-mixtures}
  (\mu *_\beta\nu)(A)
  =\int_0^\infty\int_0^\infty
    (\delta_r *_\beta\delta_s)(A)\,\mu(dr)\nu(ds).
\end{equation}
It is a commutative, associative, probability-preserving convolution,
with identity $\delta_0$. Its point-mass kernel is weakly continuous
in $(r,s)$. At $\beta=1$,
\begin{equation}
\label{eq:dimension-one-convolution}
  \delta_r *_1\delta_s
  =\tfrac12\delta_{r+s}+\tfrac12\delta_{|r-s|}.
\end{equation}

\begin{proof}
Nonnegativity under the square root follows from $|C_\beta|\leq1$.
Commutativity, preservation of total mass, and the identity property
are immediate. Weak continuity of the point-mass kernel follows by
bounded convergence against bounded continuous test functions.
For $\beta=1$, convolution is transported from ordinary convolution
of symmetric measures on $\mathbb R$ by absolute value, so it is
associative.

Suppose $\beta>1$. For a nonnegative integer $m$, write
\begin{equation}
\label{eq:moment-polynomial}
  P_m^{(\beta)}(u,v)
  =\sum_{j=0}^{\lfloor m/2\rfloor}
    \binom m{2j}(u+v)^{m-2j}(4uv)^j
    \frac{(1/2)_j}{(\beta/2)_j}.
\end{equation}
Expanding the $m$th power and using \eqref{eq:angle-moments} gives
\[
  \int z^{2m}\,(\delta_r *_\beta\delta_s)(dz)
  =P_m^{(\beta)}(r^2,s^2).
\]
For fixed $u,v$, this expression is rational in $\beta$, with no poles
on $(0,\infty)$; for fixed $\beta$, it is a polynomial in $u,v$.

Fix $r,s,t\geq0$. To compute the $2m$th moment of
$(\delta_r *_\beta\delta_s)*_\beta\delta_t$, integrate
$P_m^{(\beta)}(v,t^2)$ against the law of the squared intermediate
radius $v$. Each moment of $v$ is again given by
\eqref{eq:moment-polynomial}. Thus this iterated moment, and likewise
the moment from the other association, is rational in $\beta$.
For every integer $\beta=d\geq2$, both convolutions are norm laws
of sums of independent isotropic vectors in $\mathbb R^d$. Their
associativity follows from ordinary vector addition. Hence the difference
of the two rational moment expressions vanishes at infinitely many
integers. Its numerator is a polynomial and must vanish identically.
The two iterated convolutions therefore have equal even moments for
every $\beta>1$.

Both measures are supported on $[0,r+s+t]$. Polynomials in the squared
radius are dense in the continuous functions on that interval: apply
polynomial approximation to $u\mapsto f(\sqrt u)$ on
$[0,(r+s+t)^2]$. Equality of the even moments therefore gives equality
of the measures. This proves associativity for three point masses.
Integration against arbitrary input probability measures, using Tonelli's
theorem, proves the general assertion without moment assumptions on
the inputs.
\end{proof}

For $\beta>1$ and $r,s>0$, the support of
\eqref{eq:convolution} is $[|r-s|,r+s]$. In the standard hypergroup
terminology the involution is the identity and a Haar measure is,
up to a positive constant, $r^{\beta-1}\,dr$
\cite[Section~1]{Roesler2007}. These statements concern measures on
an ordinary half-line; a noninteger-dimensional group is not assumed.

\subsection{The Gaussian convolution semigroup}
\label{thm:gaussian-convolution}
\noindent\textbf{Theorem.}
For every $\beta\geq1$ and $s,t\geq0$,
\begin{equation}
\label{eq:gaussian-semigroup}
  \gamma_{\beta,s} *_\beta\gamma_{\beta,t}
  =\gamma_{\beta,s+t}.
\end{equation}

\begin{proof}
The case of a zero parameter follows from the identity element.
For $s,t>0$, choose independent
$R\sim\gamma_{\beta,s}$, $S\sim\gamma_{\beta,t}$, and $C_\beta$.
By Proposition~\ref{prop:splitting},
\[
  Z=SC_\beta\sim N(0,t),\qquad
  W=S\sqrt{1-C_\beta^2}\sim\gamma_{\beta-1,t}
\]
are independent of each other and of $R$. At $\beta=1$ take $W=0$.
The squared radius in \eqref{eq:convolution} is
\[
  Q=(R+Z)^2+W^2.
\]
For $\lambda\geq0$, a one-dimensional Gaussian integral gives
\[
  \mathbb E[e^{-\lambda(R+Z)^2}\mid R]
  =(1+2t\lambda)^{-1/2}
   \exp\left(-\frac{\lambda R^2}{1+2t\lambda}\right).
\]
Combining this with the gamma Laplace transforms yields
\begin{align*}
  \mathbb Ee^{-\lambda Q}
  &=(1+2t\lambda)^{-\beta/2}
    \left(1+\frac{2s\lambda}{1+2t\lambda}\right)^{-\beta/2}\\
  &=(1+2(s+t)\lambda)^{-\beta/2}.
\end{align*}
This identifies $Q$ as
$\operatorname{Gamma}(\beta/2,2(s+t))$, proving the theorem.
\end{proof}

This is Gaussian addition of variances at the level of radial laws.
Associativity of \eqref{eq:convolution-mixtures} does not supply a
single deterministic addition operation on sampled ghosts. A stochastic
kernel can be sampled, and iterated kernels define consistent radial
Markov chains; this is different from providing all Euclidean inner
products and their joint relations. There is also an elementary
multiplicative convolution, obtained by taking the law of $RS$ for
independent radii. Possessing both such operations on laws does not
by itself establish distributivity, conjugation, or matrix factorization
in a scalar algebra.

\section{Squared Bessel processes for every positive dimension}
\label{sec:bessel}

The radial process exists on a larger parameter range than the preceding
angular convolution. For every $\beta>0$ and $q\geq0$, the stochastic
differential equation
\begin{equation}
\label{eq:besq}
  dQ_t=2\sqrt{Q_t}\,dB_t+\beta\,dt,\qquad Q_0=q,
\end{equation}
has a unique nonnegative strong solution, the squared Bessel process
of dimension $\beta$; see \cite[Section~1]{PitmanWinkel2018}.
Its Laplace transform is
\begin{equation}
\label{eq:besq-laplace}
  \mathbb E_qe^{-\lambda Q_t}
  =(1+2t\lambda)^{-\beta/2}
   \exp\left(-\frac{q\lambda}{1+2t\lambda}\right),
  \qquad \lambda\geq0.
\end{equation}
One can verify this formula directly: for a fixed terminal time $T$,
substitute
\[
  u(s,x)=(1+2\lambda(T-s))^{-\beta/2}
   \exp\left(-\frac{\lambda x}{1+2\lambda(T-s)}\right)
\]
into $\partial_su+2x\partial_{xx}u+\beta\partial_xu=0$.
It\^o's formula makes $u(s,Q_s)$ a bounded local martingale and hence
a martingale. Evaluation at $s=0,T$ gives \eqref{eq:besq-laplace}.
In particular,
\begin{equation}
\label{eq:besq-from-zero}
  Q_0=0\quad\Longrightarrow\quad
  Q_t\sim\operatorname{Gamma}(\beta/2,2t),
  \qquad \sqrt{Q_t}\sim\gamma_{\beta,t}.
\end{equation}

Away from zero, the generator of $R_t=\sqrt{Q_t}$ is
\begin{equation}
\label{eq:bessel-generator}
  L_\beta f(r)
  =\frac12f''(r)+\frac{\beta-1}{2r}f'(r)
  =\frac{1}{2r^{\beta-1}}\frac{d}{dr}
       \left(r^{\beta-1}f'(r)\right).
\end{equation}
The expression on $(0,\infty)$ alone does not specify boundary behavior
at zero; construction \eqref{eq:besq} fixes the process intended here.
In particular, we do not assert an unqualified global singular-drift
SDE for $R$ when $0<\beta<1$.

For $\beta\geq1$, the transition kernels of the Bessel process can also
be obtained from $\delta_r *_\beta\gamma_{\beta,t}$; see
\cite[Section~5]{Matysiak2016}. Indeed, the Gaussian integration used
above, with $R=r$ deterministic, gives exactly the Laplace transform
\eqref{eq:besq-laplace} with $q=r^2$. For $0<\beta<1$,
\eqref{eq:besq} still defines the process, even though the angular
probability law \eqref{eq:angle-density} is unavailable.

\section{Gamma identities and genuine matrix shadows}
\label{sec:matrices}

\subsection{Dimension bookkeeping without fractional coordinates}
\label{prop:dirichlet}

For independent variables $U\sim\chi_a$ and $V\sim\chi_b$, where
$a,b>0$, gamma convolution gives
\begin{equation}
\label{eq:chi-additivity}
  U^2+V^2\sim\operatorname{Gamma}((a+b)/2,2),
  \qquad \sqrt{U^2+V^2}\sim\chi_{a+b}.
\end{equation}
This identity concerns addition of squared independent radii, not the
radial convolution of Section~\ref{sec:hypergroup}.

More generally, if $R_1,\ldots,R_n$ are independent $\chi_\beta$
variables and $S=\sum_{j=1}^nR_j^2$, then
\begin{equation}
\label{eq:dirichlet}
  \sqrt S\sim\chi_{n\beta},\qquad
  \left(\frac{R_1^2}{S},\ldots,\frac{R_n^2}{S}\right)
  \sim\operatorname{Dirichlet}(\beta/2,\ldots,\beta/2),
\end{equation}
and the displayed vector is independent of $S$.
To check this, set $y_j=sp_j$ in the product gamma density, with
$p_n=1-\sum_{j<n}p_j$. The Jacobian is $s^{n-1}$, and the transformed
density factors into a gamma density of shape $n\beta/2$ and the
Dirichlet density
\[
  \frac{\Gamma(n\beta/2)}{\Gamma(\beta/2)^n}
  \prod_{j=1}^np_j^{\beta/2-1}
\]
on the open simplex, with respect to $dp_1\cdots dp_{n-1}$.
The case $n=1$ is interpreted as the point mass at $(1)$.
These are ordinary probability laws for every $\beta>0$.

\subsection{Hermite and Laguerre matrix models}
\label{subsec:matrix-models}

The following spectral identifications are theorems of Dumitriu and
Edelman \cite[Theorems~2.12 and~3.4]{DumitriuEdelman2002}; their proofs
are not reproduced here. The scalings are specified to avoid confusing
different Gaussian conventions. In both models, diagonal indices range
over $1\leq j\leq n$, and subdiagonal indices over $1\leq j<n$.

For $n\geq1$ and $\beta>0$, let $T$ be real symmetric tridiagonal,
with independent diagonal and subdiagonal variables
\begin{equation}
\label{eq:hermite-matrix}
  T_{jj}\sim N(0,1),\qquad
  T_{j,j+1}=T_{j+1,j}\sim2^{-1/2}\chi_{\beta(n-j)}.
\end{equation}
Its eigenvalues, labeled by an independent uniform permutation, have
density on $\mathbb R^n$ proportional to
\begin{equation}
\label{eq:hermite-density}
  \exp\left(-\frac12\sum_{j=1}^n\lambda_j^2\right)
  \prod_{i<j}|\lambda_i-\lambda_j|^\beta.
\end{equation}

For $a>-1$, let $B$ be lower bidiagonal with mutually independent
nonzero entries
\begin{equation}
\label{eq:laguerre-matrix}
  B_{jj}\sim\chi_{2(a+1)+\beta(n-j)},\qquad
  B_{j+1,j}\sim\chi_{\beta(n-j)}.
\end{equation}
The eigenvalues of $BB^{\mathsf T}$, labeled in the same way, have
density on $(0,\infty)^n$ proportional to
\begin{equation}
\label{eq:laguerre-density}
  \exp\left(-\frac12\sum_{j=1}^n\lambda_j\right)
  \prod_{j=1}^n\lambda_j^a
  \prod_{i<j}|\lambda_i-\lambda_j|^\beta.
\end{equation}
All degrees of freedom in \eqref{eq:laguerre-matrix} are positive.
The parameter $a$ here is the exponent at the origin; it equals the
Laguerre parameter of \cite{DumitriuEdelman2002} minus
$1+\beta(n-1)/2$.

These are sampled matrices with ordinary real entries. Their existence
supplies spectral shadows; it neither constructs a matrix over a
fractional-dimensional scalar system nor follows from the fractal-cone
construction. A similarity-scaling identity for Hausdorff measure is
also not a general matrix change-of-variables theorem: additional
maps, measures, and Jacobian identities would have to be established.

\section{Dunkl Gaussians and effective dimension}
\label{sec:dunkl}

A different rigorous extension of Gaussian analysis is Dunkl theory.
For the type-$A_{n-1}$ permutation action on $\mathbb R^n$ with
nonnegative multiplicity $k=\beta/2$, its Gaussian measure, in the
variance convention used here, is
\begin{equation}
\label{eq:dunkl-gaussian}
  \nu_{n,\beta}(dx)
  =\frac1Z e^{-|x|^2/2}w_\beta(x)\,dx,
  \qquad w_\beta(x)=\prod_{i<j}|x_i-x_j|^\beta.
\end{equation}
This theory provides differential--difference operators, generalized
Hermite polynomials, and a positive heat semigroup
\cite[Sections~2--4]{Roesler2003}.

The effective radial dimension in \eqref{eq:dunkl-gaussian} is obtained
without any special-function argument. The weight is homogeneous of degree
\[
  h=\frac{\beta n(n-1)}2,
\]
so ordinary polar coordinates give
\begin{equation}
\label{eq:dunkl-radius}
  |X|\sim\chi_D,\qquad D=n+\frac{\beta n(n-1)}2.
\end{equation}
Indeed, writing $x=r\theta$ factors the unnormalized measure into
\[
  e^{-r^2/2}r^{n-1+h}\,dr\,w_\beta(\theta)\,d\omega(\theta),
\]
where $\omega$ is surface measure. Its angular integral is finite and
strictly positive. Thus the normalized radial factor is
$\gamma_{D,1}$. For $n=1$ the same conclusion is simply the
one-dimensional Gaussian calculation.

One can also see $D$ in the radial differential operator. Away from
the hyperplanes $x_i=x_j$, write
\[
  T_i f(x)=\partial_i f(x)+\frac\beta2\sum_{j\ne i}
        \frac{f(x)-f(s_{ij}x)}{x_i-x_j},
\]
where $s_{ij}$ exchanges coordinates $i$ and $j$. For a smooth radial
function $f(x)=g(|x|)$, direct calculation gives, for $r=|x|>0$,
\begin{equation}
\label{eq:dunkl-radial-operator}
  \sum_{i=1}^nT_i^2f(x)
  =g''(r)+\frac{D-1}{r}g'(r).
\end{equation}
To check the additional term, first $T_if=g'(r)x_i/r$. For each
$j\ne i$, exchanging $x_i,x_j$ changes this last function by
$g'(r)(x_i-x_j)/r$. The reflection contributions therefore sum to
$\beta n(n-1)g'(r)/(2r)$, in addition to the ordinary radial Laplacian.

Here the support of \eqref{eq:dunkl-gaussian} is all of $\mathbb R^n$,
which has integer Hausdorff dimension $n$. The number $D$ is a
homogeneity parameter, not the Hausdorff dimension of the support and
not a number of independent coordinates. We work on the full space
$\mathbb R^n$, including its center-of-mass direction, rather than only
the sum-zero hyperplane. The density is also the spectral density
\eqref{eq:hermite-density}; these two interpretations should not be
mistaken for a fractal scalar algebra.

\section{Gaussian fields on fractals and the final distinction}
\label{sec:fields}

Gaussian fields indexed by a fractal are yet another notion. For a
sigma-finite measure space $(F,\mu)$, an isonormal Gaussian family on
$L^2(F,\mu)$ is a centered jointly Gaussian family satisfying
\[
  \mathbb E[W(f)W(g)]=\int_F f(x)g(x)\,\mu(dx).
\]
It is not, in general, a random function $x\mapsto W(x)$. Given a
nonnegative self-adjoint operator $A$ and $s>0$, restrict to the
orthogonal complement of its kernel and define $A^{-s}$ by spectral
calculus. Then $X(f)=W(A^{-s}f)$ is a Gaussian family on the domain
of $A^{-s}$. Regularity and pointwise evaluation require additional
estimates.

Baudoin and Lacaux \cite{BaudoinLacaux2022} develop such constructions
using the Laplacian on the Sierpi\'nski gasket and related fractals.
In these constructions the fractal is the index space; a finite
collection of well-defined field values is an ordinary Gaussian vector.
This does not turn an individual value into a Gaussian scalar with
$\beta$ real components. Likewise, weighting Hausdorff measure by
$e^{-|x|^2/2}$ does not identify it with a heat-kernel measure for an
intrinsic diffusion on the fractal. An intrinsic diffusion requires a
specified Dirichlet form or generator, not merely a Hausdorff dimension.

The conclusions can now be stated without conflating the constructions.
For every $\beta>1$, Theorem~\ref{thm:cone} gives a genuine
$\beta$-dimensional fractal with a Gaussian-weighted Hausdorff measure
and the exact norm shadow $\chi_\beta$. For every $\beta\geq1$,
Sections~\ref{sec:angles}--\ref{sec:hypergroup} give a positive angular
law, a Gaussian coordinate shadow, and associative radial addition.
For every $\beta>0$, the radial gamma laws, squared Bessel processes,
and the displayed matrix models exist.

The limitations concern specified extra requirements. A metric space
of Hausdorff dimension below one cannot give an absolutely continuous
distance law. Ordinary Gaussian vector stability cannot have a
noninteger-dimensional full support in Euclidean space. Preserving
all polynomially continued Euclidean Gaussian Gram identities is
incompatible with positivity at noninteger dimension. None of these
obstructions contradicts the positive radial or spectral constructions.

Thus a fractal cone realizes the exact radial-volume component of the
idea, but not, by itself, the full ghosts-and-shadows calculus.
A stronger construction must state which observables are jointly
realized, which operations are defined, and which positivity and
consistency identities they satisfy. No claim of a complete
fractional-dimensional scalar algebra, or of novelty for the
constructions collected here, is made.

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\end{document}