A super-Brownian motion with a single point catalyst

A one-dimensional continuous measure-valued branching process is discussed, where branching occurs only at a single point catalyst described by the Dirac [delta]-function [delta]c. A (spatial) density field exists which is jointly continuous. At a fixed time t [greater-or-equal, slanted] 0, the density at z degenerates to 0 stochastically as z approaches the catalyst's position c. On the other hand, the occupation time process has a (spatial) occupation density field which is jointly continuous even at c and non-vanishing there. Moreover, the corresponding 'occupation density measure' ) at c has carrying Hausdorff-Besicovitch dimension one. Roughly speaking, density of mass arriving at c normally dies immediately, whereas creation of density mass occurs only on a singular time set. Starting initially with a unit mass concentrated at c, the total occupation time measure [infinity] equals in law a random multiple of the Lebesgue measure where that factor is just the total occupation density at the catalyst's position and has a stable distribution with index . The main analytical tool is a non-linear reaction diffusion equation (cumulant equation) in which [delta]-functions enter in three ways, namely as coefficient [delta]c of the quadratic reaction term (describing the point-catalytic medium), as Cauchy initial condition (leading to fundamental solutions and to the -density), and as external force term (related to the occupation density).