A new approach to the single point catalytic super-Brownian motion

A new approach is provided to the (critical continuous) super-Brownian motion X in R with a single point-catalyt #delta#_c as branching rate. We start from a superprocess U with constant branching rate and spatial motion given by the stable subordinator with index 1/2. We prove that the total occupation time measure #integral#_0"#infinity# ds U_s of U is distributed as the occupation density measure #lambda#"c of X at the catalyst c. This result is a superprocess analogue of the classical fact that the set of zeros of a linear Brownian motion is the range of a stable subordinator with index 1/2. We then show that the value X_t of the process X at time t is determined from the measure #lambda#"c by an explicit representation formula. On a heuristic level, this formula says that a mass #lambda#"c(ds) of 'particles' leaves the catalyst at time s and then evolves according to the Ito measure of Brownian excursions. This representation formula has important applications. First of all, with probability one, the density field x of X satisfies the heat equation outside of c with the noisy boundary condition at c given by the singularly continuous random measure #lambda#"c. In particular, x is C"#infinity# outside the catalyst. This property is in sharp contrast to the constant branching rate case. Another consequence is that the total mass X_1(R) is always strictly positive but dies out in probability as t #-># #infinity#. As a final application a new derivation of the singularity of the measure #lambda#"c is provided. (orig.)Available from TIB Hannover: RR 5549(81)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman