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 [BBMS22] 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.
References
[AS64] Handbook of mathematical functions with formulas, graphs and mathematical tables (Abramowitz, Milton; Stegun, Irene A., eds.), Applied Mathematics Series, 55, U. S. Government Printing Office, 1964 | MR | Zbl
[B03] Multifractal spectra of fragmentation processes, J. Stat. Phys., Volume 113 (2003) no. 3-4, pp. 411-430 | DOI | MR | Zbl
[BBMS22] Statistics of the maximum and the convex hull of a Brownian motion in confined geometries, J. Phys. A. Math. Theor., Volume 55 (2022) no. 14, 144002 | DOI | MR | Zbl
[Ber98] Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, 1998 | MR | Zbl
[Ber01] Homogeneous fragmentation processes, Probab. Theory Relat. Fields, Volume 121 (2001), pp. 301-318 | DOI | MR | Zbl
[Ber02] Self-similar fragmentations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 38 (2002) no. 3, pp. 319-340 | DOI | Numdam | MR | Zbl
[Ber03] The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., Volume 5 (2003) no. 4, pp. 395-416 | DOI | MR | Zbl
[BGT87] Regular variation, Encyclopedia of Mathematics and Its Applications, 27, Cambridge University Press, 1987 | DOI | MR | Zbl
[Cau32] Mémoire sur la rectification des courbes et la quadrature des surfaces courbes, Lith. de C. Mantoux, 1832
[CC11] Analytical and numerical results for first escape time in 2D, C. R. Math., Volume 349 (2011) no. 3-4, pp. 191-194 | DOI | Numdam | Zbl
[CLB09] Proof(s) of the Lamperti representation of continuous-state branching processes, Probab. Surv., Volume 6 (2009), pp. 62-89 | DOI | MR | Zbl
[DR13] Asymptotic behaviour of first passage time distributions for Lévy processes, Probab. Theory Relat. Fields, Volume 157 (2013) no. 1-2, pp. 1-45 | DOI | MR | Zbl
[DR16] Erratum to: “Asymptotic behaviour of first passage time distributions for Lévy processes”, Probab. Theory Relat. Fields, Volume 164 (2016) no. 3-4, pp. 1079-1083 | DOI | MR | Zbl
[GH10] Behavior near the extinction time in self-similar fragmentations I: The stable case, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 46 (2010) no. 2, pp. 338-368 | DOI | Numdam | MR | Zbl
[GH16] Behavior near the extinction time in self-similar fragmentations. II: Finite dislocation measures., Ann. Probab., Volume 44 (2016) no. 1, pp. 739-805 | DOI | MR | Zbl
[Haa03] Loss of mass in deterministic and random fragmentations, Stochastic Processes Appl., Volume 106 (2003) no. 2, pp. 245-277 | DOI | MR | Zbl
[Haa23] Tail asymptotics for extinction times of self-similar fragmentations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 59 (2023) no. 3, pp. 1722-1743 | DOI | MR | Zbl
[Kre08] Multifractal spectra and precise rates of decay in homogeneous fragmentations, Stochastic Processes Appl., Volume 118 (2008) no. 6, pp. 897-916 | DOI | MR | Zbl
[Kyp14] Fluctuations of Lévy processes with applications. Introductory lectures, Universitext, Springer, 2014 | DOI | MR | Zbl
[Lam67a] Continuous state branching process, Bull. Am. Math. Soc., Volume 73 (1967), pp. 382-386 | DOI | MR | Zbl
[Lam67b] On random time substitutions and the Feller property, Markov Processes and Potential Theory (Proc Sympos. Math. Res. Center, Madison, Wis., 1967) (1967), pp. 87-101 | MR | Zbl
[Let93] An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk, J. Theor. Probab., Volume 6 (1993) no. 2, pp. 385-387 | DOI | MR | Zbl
[RBGV14] Exit Time Distribution in Spherically Symmetric Two-Dimensional Domains, J. Stat. Phys., Volume 158 (2014) no. 1, pp. 192-230 | DOI | MR | Zbl
[SSH07] The narrow escape problem for diffusion in cellular microdomains, Proc. Natl. Acad. Sci. USA, Volume 104 (2007) no. 41, pp. 16098-16103 | DOI
[SSHE06] Narrow escape. I, J. Stat. Phys., Volume 122 (2006) no. 3, pp. 437-463 | DOI | MR | Zbl
[Tem90] Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions, SIAM J. Math. Anal., Volume 21 (1990) no. 1, pp. 241-261 | DOI | MR | Zbl
[TV17] Brunn–Minkowski theory and Cauchy’s surface area formula, Am. Math. Mon., Volume 124 (2017) no. 10, pp. 922-929 | DOI | MR | Zbl