Searcharxiv⌕ 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

Loglinear Hawkes processes: existence, nonexplosion, stability and moment bounds

This paper discusses a special class of nonlinear Hawkes processes, where the rate function is the exponential function. We call these processes loglinear Hawkes processes. Such processes are used with success in computational neuroscience and we provide a mathematical background for them. After obtaining existence from a standard Poisson embedding, we give sufficient conditions for explosion and nonexplosion that cover a large class of memory functions. In order to give sufficient stability conditions, we construct an autoregressive integer-valued time series that gives pathwise upper bounds for the bin counts of a loglinear Hawkes process. From here, a Markov chain argument shows that the regeneration time for a loglinear Hawkes process has finite expectation if the memory function is negative enough in a neighborhood of zero. Stability in total variation follows. Finally, we show that both the loglinear Hawkes process under a large class of initial conditions and its stationary version have finite exponential moments.

math.PR↗
arXiv2026

Eddy viscosity by Lévy transport noises

We consider stochastic linear transport equations in dimension $d\geq 2$ and stochastic 2D Euler equations with $L^2$-initial condition, both driven by Lévy transport noise in the Marcus sense. Under a suitable scaling limit of the noises, we prove that the solutions to the stochastic linear transport equations converge to the solution of the deterministic heat equation, while weak solutions to the stochastic 2D Euler equations converge weakly to the unique solution of the deterministic 2D Navier--Stokes equation. These results show that small-scale jump noises generate eddy viscosity, extending the recent studies on Itô--Stratonovich diffusion limits to the discontinuous setting.

math.PR↗
arXiv2026

Triggered urn models for frequently asked questions (FAQ)

We investigate a nonclassic urn model with triggers that increase the number of colors. The scheme has emerged as a model for web services that set up frequently asked questions (FAQ). We present a thorough asymptotic analysis of the FAQ urn scheme in generality that covers a large number of special cases, such as Simon's urn. For instance, we consider time dependent triggering probabilities. We identify regularity conditions on these probabilities that classify the schemes into those where the number of colors in the urn remains almost surely finite or increases to infinity and conditions that tell us whether all the existing colors are observed infinitely often or not. We determine the rank curve, too. In view of the broad generality of the trigger probabilities, a spectrum of limit distributions appears, from central limit theorems to Poisson approximation, to power-laws, revealing connections to Heaps exponent and Zipf's law. A combinatorial approach to the Simon urn is presented to indicate the possibility of such exact analysis, which is important for short-term predictions. Extensive simulations on real datasets (from Amazon sales) as well as computer-generated data clearly indicate that the asymptotic and exact theory developed agrees with practice.

math.PR↗