A new approach to the single point catalytic super-Brownian motion
- Fleischmann, Klaus
- Le Gall, Jean-Francois
2010 Mathematics Subject Classification
- 60J80 60J55 60G57
- Super-stable subordinator, sample path smoothness, Campbell measure formula, canonical measures, backward measurability, point-catalytic medium, critical branching, super-Brownian motion, superprocess, measure-valued branching process, total extinction
A new approach is provided to the (critical continuous) super-Brownian motion X in R with a single point-catalyst δc 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 ∫0∞ ds Us of U is distributed as the occupation density measure λc 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 Xt of the process X at time t is determined from the measure λc by an explicit representation formula. On a heuristic level, this formula says that a mass λc(ds) of ''particles'' leaves the catalyst at time s and then evolves according to the Itô 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 λc. In particular, x is C∞ outside the catalyst. This property is in sharp contrast to the constant branching rate case. Another consequence is that the total mass Xt(R) is always strictly positive but dies out in probability as t → ∞. As a final application a new derivation of the singularity of the measure λc is provided.
- Probab. Theory Relat. Fields 102, 63-82 (1995)