A super-Brownian motion with a single point catalyst

Abstract

A one-dimensional continuous measure-valued branching process { H t;t ⩾ } is discussed, where branching occurs only at a single point catalyst described by the Dirac δ-function δ c . A (spatial) density field { x t(z); t ⩾ 0, z ≠ c} exists which is jointly continuous. At a fixed time t ⩾ 0, the density x t(z) at z degenerates to 0 stochastically as z approaches the catalyst's position c. On the other hand, the occupation time process Y t ≔ ∫ t 0 dr H r(·) has a (spatial) occupation density field { y t(z); t ⩾ 0, z ϵ } which is jointly continuous even at c and non-vanishing there. Moreover, the corresponding ‘occupation density measure’ d y t(c) ≔ λ c(dt ) 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 Y ∞ 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 1 2 . The main analytical tool is a non-linear reaction diffusion equation (cumulant equation) in which δ-functions enter in three ways, namely as coefficient δ c of the quadratic reaction term (describing the point-catalytic medium), as Cauchy initial condition (leading to fundamental solutions and to the x -density), and as external force term (related to the occupation density).