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August 1996 A general stochastic model for nucleation and linear growth
L. Holst, M. P. Quine, J. Robinson
Ann. Appl. Probab. 6(3): 903-921 (August 1996). DOI: 10.1214/aoap/1034968233


The model considered here has arisen in a number of completely separate contexts: release of neurotransmitter at neuromuscular synapses, unravelling of strands of DNA, differentiation of cells into heterocysts in algae and growth of crystals. After a shear transformation the model becomes a Markov process, based on a Poisson process on the upper half plane, homogeneous in the horizontal (time) direction, which increases at unit rate except for occasional "drops." By considering the process separately when it is above or below a given "level," for any interval on the time axis, we obtain in particular exact moment results and prove asymptotic normality for long time intervals for the number of downcrossings in the interval, the total time in the interval when the process is below the specified level and the number of drops in the interval. Limit distributions as the length of interval tends to infinity are obtained for the level at which the interval is "covered." It is shown that several problems considered in the literature have analytic solutions as special cases of the general model. The numerical results from one special case are compared to statistics obtained from experimental data from neurobiology.


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L. Holst. M. P. Quine. J. Robinson. "A general stochastic model for nucleation and linear growth." Ann. Appl. Probab. 6 (3) 903 - 921, August 1996.


Published: August 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0860.60059
MathSciNet: MR1410121
Digital Object Identifier: 10.1214/aoap/1034968233

Primary: 60J15 , 60J25 , 60K15 , 60K40

Keywords: cell biology , crystal aggregates , DNA , inhibition , nucleation , Piecewise deterministic Markov processes , shear transformation , Synapses

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 3 • August 1996
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