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How fast are the particles of super-Brownian motion?
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  • Published: October 2001

How fast are the particles of super-Brownian motion?

  • Peter Mörters1 

Probability Theory and Related Fields volume 121, pages 171–197 (2001)Cite this article

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Abstract.

In this paper we investigate fast particles in the range and support ofsuper-Brownian motion in the historical setting. In this setting eachparticle of super-Brownian motion alive at time t is represented by apath w:[0,t]→ℝd and the state of historical super-Brownian motionis a measure on the set of paths. Typical particles have Brownian paths,however in the uncountable collection of particles in the range of asuper-Brownian motion there are some which at exceptional times movefaster than Brownian motion. We determine the maximal speed of allparticles during a given time period E, which turns out to be afunction of the packing dimension of E. A path w in the support ofhistorical super-Brownian motion at time t is called a-fast if . Wecalculate the Hausdorff dimension of the set of a-fast paths in thesupport and the range of historical super-Brownian motion. A valuabletool in the proofs is a uniform dimension formula for the Browniansnake, which reduces dimension problems in the space of stopped paths to dimension problems on the line.

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Authors and Affiliations

  1. Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67663 Kaiserslautern, Germany. e-mail: peter@mathematik.uni-kl.de, , , , , , DE

    Peter Mörters

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  1. Peter Mörters
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Additional information

Received: 27 January 2000 / Revised version: 28 August 2000 / Published online: 24 July 2001

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Mörters, P. How fast are the particles of super-Brownian motion?. Probab Theory Relat Fields 121, 171–197 (2001). https://doi.org/10.1007/PL00008801

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  • Issue Date: October 2001

  • DOI: https://doi.org/10.1007/PL00008801

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Keywords

  • Brownian Motion
  • Maximal Speed
  • Hausdorff Dimension
  • Typical Particle
  • Dimension Problem
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