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Superprocesses in random environments. (English) Zbl 0874.60041

The paper establishes the weak convergence of a sequence of branching particle systems whose branching mechanism is affected by a random environment. The environment is specified by a sequence \(\xi_k\), \(k=1,2,\dots\), of independent, identically distributed zero-mean random fields on the space \(E\) in which the particles live. The \(n\)th particle system \(X^n_t\), \(t\geq 0\), starts at time \(t=0\) with \(K_n\) particles \((K_n\sim n)\) and during any time interval \((k/n,(k+1)/n)\), \(k=0,1,2,\dots\), all existing particles move independently of each other in accordance with a Feller process on \(E\). Conditionally on \(\xi^{(n)}_k\) (where \(k\geq 1\) and \(\xi^{(n)}_k\) is \(\xi_k\) truncated at \(\pm\sqrt n\)), each particle existing at time \(k/n\) either splits into two particles or dies, with probabilities \({1\over 2}+{1\over 2\sqrt n} \xi^{(n)}_k(x)\) and \({1\over 2}-{1\over 2\sqrt n} \xi^{(n)}_k(x)\), respectively, independently of other particles. Here \(x\) denotes the location of the particle. The author proves that if \(X^n_0\) converges, then the sequence \(X^n\), \(n=1,2,\dots\), converges weakly to a measure-valued superprocess which is the unique solution of a certain martingale problem.

MSC:

60G57 Random measures
60F17 Functional limit theorems; invariance principles
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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