1994
A superBrownian motion with a single point catalyst
Publication
Publication
Stochastic Processes and their Applications , Volume 49  Issue 1 p. 3 40
A onedimensional continuous measurevalued branching process {Ht;t ≥ } is discussed, where branching occurs only at a single point catalyst described by the Dirac δfunction 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 nonvanishing there. Moreover, the corresponding 'occupation density measure' dyt(c) {colon equals} λc(dt) at c has carrying HausdorffBesicovitch 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 nonlinear reaction diffusion equation (cumulant equation) in which δfunctions enter in three ways, namely as coefficient δc of the quadratic reaction term (describing the pointcatalytic medium), as Cauchy initial condition (leading to fundamental solutions and to the xdensity), and as external force term (related to the occupation density).
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doi.org/10.1016/03044149(94)901104  
Stochastic Processes and their Applications  
Organisation  Carleton University 
Dawson, D.A, & Fleischmann, K. (Klaus). (1994). A superBrownian motion with a single point catalyst. Stochastic Processes and their Applications, 49(1), 3–40. doi:10.1016/03044149(94)901104
