WIAS Preprint No. 197, (1995)

A super-Brownian motion with a locally infinite catalytic mass


Authors

  • Fleischmann, Klaus
  • Mueller, Carl

2010 Mathematics Subject Classification

  • 60J80 60J55 60G57

Keywords

  • hyperbolic branching rate, strong killing, infinite point catalyst, Feynman-Kac equation, killed Brownian motion, historical process, super-Brownian motion, superprocess, branching functional of infinite (local) characteristic, measure-valued branching, catalytic superprocess

DOI

10.20347/WIAS.PREPRINT.197

Abstract

A super-Brownian motion X in ℝ with "hyperbolic" branching rate ρ2 (b) = 1/b2, b ∈ ℝ, is constructed, which symbolically could be described by the formal stochastic equation

dXt = ½ ΔXt dt + √2ρ2XtdWt,   t ≥ 0,  (1)

(with a space-time white noise W).

If the finite starting measure X0 does not have mass at b = 0, then this superprocess X will never hit the catalytic center: There is Brownian stopping time r strictly smaller than the hitting time of 0 such that Dynkin's stopped measures Xr vanishes except a.s.

Appeared in

  • Probab. Theory Related Fields, 107 (1997), pp. 325-357

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