Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16339

Distinguishing sets and measures from their planar Brownian images

Sascha Troscheit
View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivated by a question of Hochman, we investigate whether two deterministic time measures or time sets can be distinguished almost surely from a single planar Brownian image. Using rescaled correlation integrals, we give a criterion for mutual singularity of the laws of the image measures, even when the time measures have the same dimension. For self-similar time sets satisfying the open set condition, we also recover the natural occupation measure from the compact image. Together, these results allow us to distinguish lattice and non-lattice self-similar sets and their natural measures within this class. For a naturally parametrised family with two contractions, with fixed dimension, each parameter can be distinguished from all but finitely many others. If the dimension is sufficiently small, every distinct parameter can be distinguished.

Explore related subjects

math.PRmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sascha Troscheit. 2026-10-12. Distinguishing sets and measures from their planar Brownian images. https://arxiv.org/abs/2610.16339

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Free probability via entropic optimal transport

Let $μ$ and $ν$ be compactly supported probability measures on $\mathbb{R}$, and let $μ\boxplus ν$ denote their additive free convolution. We show that for all sufficiently large real $z$, \begin{equation} \int_{-\infty}^\infty \log(z - x) \, (μ\boxplus ν)(\mathrm{d}x) = \sup_Π \left\{ \mathbf{E}_Π[\log(z - (X+Y))] - H(Π\mid μ\otimes ν) \right\}, \end{equation} where the supremum is taken over all couplings $Π$ of $μ$ and $ν$. Analogous formulas hold for multiplicative free convolution $μ\boxtimes ν$ and free compression $[μ]_τ$. In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer $Π_z$ encodes the subordination equations via couplings of random variables. Our approach is based on a large deviation principle on the symmetric group, combined with the quadrature method of Marcus--Spielman--Srivastava.

math.PR↗
arXiv2026

Scaling limit of boundary random walks: A martingale problem approach

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the $J_1$-Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.

math.PR↗
arXiv2026

Random Polyhedral Cones I: Distributional Results via Gale Duality

Let $U_1,\ldots,U_n$ be independent random vectors uniformly distributed on the unit sphere $\mathbb S^{d-1}\subseteq\mathbb R^d$, where $n\ge d$, and consider the random polyhedral cone \[ \mathcal W_{n,d}:=\mathop{\mathrm{pos}} (U_1,\ldots,U_n) = \{λ_1 U_1+ \ldots + λ_n U_n: λ_1\geq 0, \ldots, λ_n \geq 0\}. \] We establish several distributional results for $\mathcal W_{n,d}$ and the associated spherical polytope $\mathcal W_{n,d}\cap\mathbb S^{d-1}$. Our main contributions include: (i) Let $α_d$ denote the solid angle of $\mathcal W_{d,d}$ and write $m(d,k):=\mathbb E[α_d^k]$ for its $k$-th moment. We prove the symmetry $m(d,k)=m(k,d)$. As an application, we compute $\mathop{\mathrm{Var}}[α_d]=2^{-d}(d+1)^{-1}-4^{-d}$ and derive a closed formula for the third moment. (ii) For $n=d+1,d+2,d+3$ we determine the probability that $\mathcal W_{n,d}\cap\mathbb S^{d-1}$ is a spherical simplex, a spherical analogue of the classical Sylvester problem. In the case $n=d+2$ we also determine the distribution of the number of vertices of $\mathcal W_{d+2,d}\cap\mathbb S^{d-1}$. (iii) Let $f_\ell(\mathcal W_{n,d})$ denote the number of $\ell$-dimensional faces of $\mathcal W_{n,d}$. We prove a distributional limit theorem for $f_\ell(\mathcal W_{n,d})$ in the regime $n=d+k$ and $\ell=d-q$, where $k,q\in\mathbb N$ are fixed and $d\to\infty$. The limit law is a weighted sum of independent chi squared variables, with weights given by explicit eigenvalues of a convolution operator on the sphere. A unifying ingredient is an explicit coupling producing i.i.d. uniform vectors $U_1,\ldots,U_n\in\mathbb S^{d-1}$ together with i.i.d. uniform vectors $V_1,\ldots,V_n\in\mathbb S^{n-d-1}$ whose associated oriented matroids are Gale dual.

math.PR↗