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Absolute continuity of the measure states in a branching model with catalysts. (English) Zbl 0724.60089

Stochastic processes, Proc. Semin., Vancouver/Can. 1990, Prog. Probab. 24, 117-160 (1991).
[For the entire collection see Zbl 0716.00012.]
A critical superprocess is discussed in a one-dimensional random medium fluctuating both in time and space. Typically, the medium is given by a dense system of independently moving random point catalysts. Branching occurs only in the presence of these catalysts. The main result says that despite this highly singular varying medium the states of that superprocess with motion index \(\alpha\) and branching index \(1+\beta\) (both in the interval (1,2]) have density functions (a.s.). The process was constructed by D. A. Dawson and K. Fleischmann [Critical branching in a highly fluctuating random medium, Probab. Theory Relat. Fields (1991; to appear)]. The key approach consists in showing that the (nonlinear) cumulant equation in the random medium, describing the log- Laplace transition functional of the process, has (mild) basic solutions. That equation was first studied by D. A. Dawson and K. Fleischmann [Diffusion and reaction caused by point catalysts, SIAM J. Appl. Math. (to appear)].

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G57 Random measures
60K35 Interacting random processes; statistical mechanics type models; percolation theory
58J35 Heat and other parabolic equation methods for PDEs on manifolds
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Citations:

Zbl 0716.00012