WIAS Preprint No. 450, (1998)

On the hot spots of a catalytic super-Brownian motion



Authors

  • Delmas, Jean-François
  • Fleischmann, Klaus

2010 Mathematics Subject Classification

  • 60J80 60G57 60K35

Keywords

  • Catalytic super-Brownian medium, superprocess, measure-valued process, collision local time of catalyst and reactant, two-dimensional process, catalytic medium

Abstract

Consider the catalytic super-Brownian motion X𝜚 (reactant) in ℝd, d ≤ 3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion 𝜚 (catalyst). Our main object of study is the collision local time L = L[𝜚,X𝜚](d[s,x]) of catalyst and reactant. It determines the covariance measure in the martingale problem for X𝜚 and reflects the occurence of "hot spots" of reactant which can be seen in simulations of X𝜚. In dimension 2, spatial marginal collision densities exist and, via self-similarity, enter as factor in the long-term random ergodic limit of L (diffusiveness of the 2-dimensional model).

Appeared in

  • Probab. Theory Relat. Fields, 121(3) (2001), pp. 389-421

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