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.
Author information
Authors and Affiliations
Additional information
Received: 27 January 2000 / Revised version: 28 August 2000 / Published online: 24 July 2001
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1007/PL00008801
Keywords
- Brownian Motion
- Maximal Speed
- Hausdorff Dimension
- Typical Particle
- Dimension Problem