Electronic Communications in Probability

On the volume of the shrinking branching Brownian sausage

Mehmet Öz

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Abstract

The branching Brownian sausage in $\mathbb{R} ^{d}$ was defined in [4] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R} ^{d}$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated branching Brownian sausage (BBM-sausage) with radius exponentially shrinking in time. Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.

Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 37, 12 pp.

Dates
Received: 6 December 2019
Accepted: 3 May 2020
First available in Project Euclid: 13 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1589335619

Digital Object Identifier
doi:10.1214/20-ECP316

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F15: Strong theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 92D25: Population dynamics (general)

Keywords
branching Brownian motion density sausage strong law of large numbers

Rights
Creative Commons Attribution 4.0 International License.

Citation

Öz, Mehmet. On the volume of the shrinking branching Brownian sausage. Electron. Commun. Probab. 25 (2020), paper no. 37, 12 pp. doi:10.1214/20-ECP316. https://projecteuclid.org/euclid.ecp/1589335619


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References

  • [1] Bolthausen, E.: On the volume of the Wiener sausage. Ann. Probab. 18(4), (1992), 1576 – 1582.
  • [2] Bramson, M.: Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31(5), (1978), 531 – 581.
  • [3] Donsker, M. D. and Varadhan, S. R. S.: Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28(4), (1975), 525 – 565.
  • [4] Engländer, J.: On the volume of the supercritical super-Brownian sausage conditioned on survival. Stochastic Process. Appl. 88, (2000), 225 – 243.
  • [5] Hamana, Y.: On the expected volume of the Wiener sausage. J. Math Soc. Jpn. 62(4), (2010), 1113 – 1136.
  • [6] Hamana, Y. and Matsumoto, H.: The probability distributions of the first hitting times of Bessel processes. Trans. Amer. Math Soc. 365(10), (2013), 5237 – 5257.
  • [7] Karlin, S. and Taylor, M.: A First Course in Stochastic Processes. Second edition. Academic Press, New York-London, 1975. xvii+557 pp.
  • [8] Le Gall, J.-F.: Sur une conjecture de M. Kac. Probab. Theory Relat. Fields 78(3), (1988), 389 – 402.
  • [9] Öz, M., Çağlar, M. and Engländer, J.: Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles. Ann. Inst. H. Poincaré Probab. Statis. 53(2), (2017), 842 – 864.
  • [10] Öz, M.: Large deviations for local mass of branching Brownian motion, arXiv:1811.09037
  • [11] Öz, M.: On the density of branching Brownian motion in subcritical balls, arXiv:1909.06197
  • [12] Spitzer, F.: Some theorems concerning $2$-dimensional Brownian motion. Trans. Amer. Math. Soc. 87(1), (1958), 187 – 197.
  • [13] Spitzer, F.: Electrostatic Capacity, Heat Flow, and Brownian Motion. Z. Wahrsch. und Verw. Gebiete 3(2), (1964), 110 – 121.
  • [14] Sznitman, A.-S.: Long time asymptotics for the shrinking wiener sausage. Comm. Pure Appl. Math. 43(6), (1990), 809 – 820.