Mutually catalytic branching in the plane: Infinite measure states
The authors construct a pair of infinite-measure-valued processes as a model for two-type population in \(\mathbb R^{2}\) which undergoes diffusion and branching. It is a rescaled limit of a corresponding \(\mathbb Z^{2}\)-lattice model studied by \textit{D. A. Dawson} and \textit{E. A. Perkins} [Ann. Probab. 26, 1088-1138 (1998; Zbl 0938.60042)]. The branching rate of each type of population is proportional to the local density of the other type, whereas the collision rate is sufficiently small compared with the diffusion rate. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of two populations. Local extinction of one type is shown (in law in the long-term limit) and the surviving population is uniform with random intensity. Uniqueness questions are treated by \textit{D. A. Dawson}, \textit{K. Fleischmann}, \textit{L. Mytnik}, \textit{E. A. Perkins} and \textit{J. Xiong} [Ann. Inst. Henri Poincaré, Probab. Stat. 39, 135-191 (2003; Zbl 1016.60091)].
interactive and state-dependent branching
scientific article; zbMATH DE number 1868594