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.
@article{1034968233,
author = {Holst, L. and Quine, M. P. and Robinson, J.},
title = {A general stochastic model for nucleation and linear
growth},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 903-921},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968233}
}
Holst, L.; Quine, M. P.; Robinson, J. A general stochastic model for nucleation and linear
growth. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 903-921. http://gdmltest.u-ga.fr/item/1034968233/