WIAS Preprint No. 510, (1999)

Catalytic and mutually catalytic branching



Authors

  • Dawson, Donald A.
  • Fleischmann, Klaus

2010 Mathematics Subject Classification

  • 60K35 60J80 60G57

Keywords

  • Catalytic super-Brownian motion, catalytic super-random walk, catalyst, reactant, superprocess, measure-valued branching, absolute continuity, self-similarity, collision local time, glycolysis, martingale problem, segregation of types, coexistence of types, self-duality

Abstract

In this paper we present an overview of recent work on lattice and measure-valued models of catalytic reactions and in particular catalytic branching systems. The main phenomena exhibited by nearly critical branching systems are dimension-dependent clumping in small and large space and time scales. Special emphasis is given to the effects which occur when the catalyst is highly clumped and in particular when in the continuum models the catalyst is a time-dependent singular measure. Finally, the interactive model of mutually catalytic branching is described and some recent results are reviewed. - The basic tools include log-Laplace functionals, measure-valued martingale problems, collision local times, and duality.

Appeared in

  • Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natururkd. 1. Reeks. K. Ned. Akad. Wet., 52, R. Neth. Acad. Arts Sci., Amsterdam 2000

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