Multiple scale analysis of spatial branching processes under the Palm distribution
The paper deals with the continuous-time critical binary branching random walk on \(\mathbb{Z}^2\) where the random walk kernel is assumed to have a finite second moment, and with its continuous mass analogue, a linearly interacting critical Feller's branching diffusion on \(\mathbb{Z}^2\) (a particular super-random walk). Via log-Laplace transition functionals, both models have the same deterministic dual process, the solution of a ``diffusion''-reaction (or log-Laplace) equation on \(\mathbb{Z}^2\) with quadratic reaction term. Two well-known concepts are used to describe the formation of clusters in detail: (i) the Palm distribution approach, and (ii) multiple space-time scaling. It is shown that both approaches commute. Technically, this leads to tree indexed branching systems which are completely characterized by their time-space moments. These moments are described effectively and explicitly.
grove indexed systems of branching models
scientific article; zbMATH DE number 1868592