On soft and hard particle motions for stochastic marked point processes

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This article analyses the space–time interdependency of marked point processes considering marked point random and deterministic motions. In this context, marked points can move randomly or/and deterministically as a result of their own growing status or/and the growing status of their neighbours. A continuous stochastic process for the generation of space–time patterns of moving marked points is formulated and illustrated. Three deterministic models of collision of moving and growing hard particles are also developed. We consider deterministic particle motion promoted by: (a) particle's own growth when touching other neighbours; (b) the growth of touching neighbours; and (c) a combination of these two collision dynamics. Moreover, we extend this combined deterministic model into its counterpart stochastic approach. Our analysis suggests that the generating mechanism of the space–time stochastic process may not become fully reflected in snap-shot analyses of purely spatial patterns when assuming random motion of marked points. Thus, the understanding of such complex dynamics may involve not only the analysis of purely spatial patterns but also their interaction in the space–time domain.