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Trimmed trees and embedded particle systems. (English) Zbl 1048.60063

A superprocess \({\mathcal X}=({\mathcal X}_t)_{t\geq 0}\) taking on values in the space \({\mathcal M}(E)\) of finite measures on some space \(E\) can be rather often related with a branching particle process \(X=(X_t)_{t\geq 0}\) by the formula \[ P^{{\mathcal L}(\text{Pois} (\mu))}[X_t\in\cdot]=P^\mu[ \text{Pois} ({\mathcal X}_t)\in\cdot],\quad t\geq 0,\,\mu\in{\mathcal M}(E), \] where \(\text{Pois}({\mathcal X}_t)\) denotes a Poisson point measure with random intensity \({\mathcal X}_t\) and \(P^{{\mathcal L}(\text{Pois} (\mu))} \) denotes the law of the process \(X\) started with the initial law \({\mathcal L}( \text{Pois} (\mu)).\) This relation holds, for instance, when \({\mathcal X}\) is the standard critical continuous super-Brownian motion in \(R^d\) which corresponds to the evolution equation \(\frac{{\partial}}{{\partial} t}u_t=\frac{1}{2} \Delta u_t-u_t^2\) and \(X\) is a system of binary Brownian motions with branching rate 1 and death rate 1. One can say that \(X\) can be obtained from \({\mathcal X}\) by Poissonization.
The authors investigate Poissonization relations for a class of continuous superprocesses on compacta with Feller underlying motion and describe conditions under which a superprocess \({\mathcal X}\) and a branching particle system \(X\) can be coupled as processes in such a way that \[ P[X_t\in\cdot\mid ({\mathcal X}_s)_{0\leq s\leq t}]=P[\text{Pois} (h{\mathcal X}_t)\in\cdot\mid {\mathcal X}_t] \text{ a.s. }\forall t\geq 0, \] where \(h\) is a sufficiently smooth density.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G57 Random measures
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