A one-dimensional continuous measure-valued branching process {Ht;t ≥ } is discussed, where branching occurs only at a single point catalyst described by the Dirac δ-function δc. A (spatial) density field {xt(z); t ≥ 0, z ≠ c} exists which is jointly continuous. At a fixed time t ≥ 0, the density xt(z) at z degenerates to 0 stochastically as z approaches the catalyst's position c. On the other hand, the occupation time process Yt {colon equals} ∫t 0 dr H r(·) has a (spatial) occupation density field {yt(z); t ≥ 0, z ε{lunate} } which is jointly continuous even at c and non-vanishing there. Moreover, the corresponding 'occupation density measure' dyt(c) {colon equals} λc(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 δc 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).

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Stochastic Processes and their Applications
Carleton University

Dawson, D.A, & Fleischmann, K. (Klaus). (1994). A super-Brownian motion with a single point catalyst. Stochastic Processes and their Applications, 49(1), 3–40. doi:10.1016/0304-4149(94)90110-4