Fragmentation processes and the convex hull of the Brownian motion in a disk

Bénédicte Haas; Bastien Mallein

Home/Archive/arXiv

arXiv · 2312.01952

Fragmentation processes and the convex hull of the Brownian motion in a disk

Bénédicte Haas·Bastien Mallein

View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivated by the study of the convex hull of the trajectory of a Brownian motion in the unit disk reflected orthogonally at its boundary, we study inhomogeneous fragmentation processes in which particles of mass $m \in (0,1)$ split at a rate proportional to $|\log m|^{-1}$. These processes do not belong to the well-studied family of self-similar fragmentation processes. Our main results characterize the Laplace transform of the typical fragment of such a process, at any time, and its large time behavior. We connect this asymptotic behavior to the prediction obtained by physicists in \cite{DBBM22} for the growth of the perimeter of the convex hull of a Brownian motion in the disc reflected at its boundary. We also describe the large time asymptotic behavior of the whole fragmentation process. In order to implement our results, we make a detailed study of a time-changed subordinator, which may be of independent interest.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Bénédicte Haas, Bastien Mallein. 2023-12-04. Fragmentation processes and the convex hull of the Brownian motion in a disk. https://arxiv.org/abs/2312.01952

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

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗

arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗