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Survival functions and contact distribution functions for inhomogeneous, stochastic geometric marked point processes. (English) Zbl 1068.60016

A birth-and-growth process is composed of two processes: birth (nucleation) and subsequent growth of spatial cells (crystals). The processes are generally stochastic both in time and space, meaning that the nuclei are born and grow up at random places and at random moments. Such a process can be used as a mathematical model for phenomena like tumor growth, forest growth and so on. The authors analyze these processes by methods of survival analysis, extending to the spatially heterogeneous case partial results known for homogeneous Poisson nucleation processes. They give general expressions for the survival and the hazard functions, and a link between them. Here, a survival function \(S(t,x)\) of a point \(x\) at time \(t\) represents the probability that \(x\) is not crystallized at time \(t\), and the hazard function \(h(t,x)\) represents the rate of capture of the point \(x\). The authors also obtain a relation between the hazard function and the contact distribution function of stochastic geometry.

MSC:

60D05 Geometric probability and stochastic geometry
28A75 Length, area, volume, other geometric measure theory
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References:

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