On Random and Gibbsian Particle Motions for Point Processes Evolving in Space and Time

Abstract

This article presents an analysis of space-time interdependencies of spatial point processes considering random and deterministic Gibbsian point motions caused by repulsion effects between particles. Two deterministic models of Gibbsian motions are considered by formulating a constant (i.e., Strauss-like) and a linear interaction motion functions. Given that theoretical development of continuous space-time stochastic processes are mathematically intractable, we have mainly based our analysis on numerical simulations. Our results suggest that to fully understand such complex dynamics, the analysis of purely spatial patterns should be combined with their interactions in the space-time domain. Otherwise, analysis of pure spacial patterns may not fully explain the real mechanism generating such dynamical configurations. We highlight that adding movement to sedentary points opens new areas of application and research to study biological phenomena, where particles not only evolve through time but also can change spatial positions in terms of their neighbor locations.

Keywords:

• Deterministic and stochastic particle motions
• Gibbs processes
• Point processes
• Space-time interactions
• Strauss-like interaction functions

Mathematics Subject Classification:

• 60G55
• 60K35
• 62H11
• 62M30

Acknowledgments

The Editor and referees are acknowledged with thanks. Their precise comments and suggestions have clearly improved an earlier version of the manuscript.