| Abstract: |
| Interacting particle systems have been widely used in many applied science fields, such as social science, chemistry, biology, physics, etc. Many interacting particle systems are Markov pure jump processes and a martingale representation can be derived for the system. The martingale representation can then be used to derive the hydrodynamic limit of the systems. The analogue of this methodology is the law of large numbers. Moreover, the fluctuation of the system deviating from its hydrodynamic limit can also be derived.
The above theoretical framework is applied by us to the agent-based model of crime behavior in urban area. The result is a quantitative and applicable stochastic-statistical model. Particularly, we have made progress to the pioneering statistical agent-based model of residential burglary (Short et al., Math. Models Methods Appl., 2008) in two space dimensions. That is, we use the Poisson clocks to govern the time steps of the evolution of the model, rather than deterministic time increments used in the previous works. Poisson clocks are particularly suitable to model the times at which random events arrive. Introduction of Poisson clock not only produces similar simulation output, but also brings in theoretically the martingale approach mentioned above. The hydrodynamic limit is considered for the first time for this model and an indication that it may coincide with a well-known mean-field continuum limit is found. Also the stochastic fluctuation of the system is studied and analyzed quantitatively. This is the first time such things are done to this model. |
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