Electronic Communications in Probability

On the volume of the shrinking branching Brownian sausage

Mehmet Öz

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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

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

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

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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)

branching Brownian motion density sausage strong law of large numbers

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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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  • [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.