相互作用粒子系统的过去与模糊和扰动的耦合
Coupling from the past times with ambiguities and perturbations of interacting particle systems
摘要：
我们讨论了耦合从过去的技术（CFTP）的扰动的相互作用的粒子系统的d维整数格，与一个有限集的状态，在框架内的图形建设的动态泊松过程的基础上。我们首先开发的一般结果，我们称之为CFTP时间的模糊性。这些是类似于经典的耦合（从过去）的时间，除了耦合属性只提供了一些模糊的随机演化的系统得到解决。如果这些模糊性平均而言足够罕见，则具有模糊性的CFTP时间可以用于构建实际CFTP时间，其属性可以根据具有模糊性的原始CFTP时间的属性来控制。然后，我们证明了一个一般的扰动结果，它可以非正式地陈述如下。从一个具有CFTP时间的相互作用粒子系统开始，其定义涉及图形结构中指数可积点数的探索，并且满足正速率性质。然后考虑通过向原始动力学添加新的跃迁而获得的扰动。我们的结果表明，只要扰动足够小（在足够小的速率的意义上），扰动的相互作用粒子系统也具有CFTP时间（具有良好的性质，如指数衰减的尾巴）。证明包括在定义一个CFTP时间与模糊性的扰动动力学，从CFTP时间的未扰动动力学。最后，我们讨论了粒子系统的例子，这一结果可以应用。具体的例子包括一类邻居依赖的核苷酸取代模型，和经典的选举模型的变化，说明我们的方法超越弱相互作用的粒子系统的情况下的能力。
英文摘要：
We discuss coupling from the past techniques (CFTP) for perturbations of interacting particle systems on the d-dimensional integer lattice, with a finite set of states, within the framework of the graphical construction of the dynamics based on Poisson processes. We first develop general results for what we call CFTP times with ambiguities. These are analogous to classical coupling (from the past) times, except that the coupling property holds only provided that some ambiguities concerning the stochastic evolution of the system are resolved. If these ambiguities are rare enough on average, CFTP times with ambiguities can be used to build actual CFTP times, whose properties can be controlled in terms of those of the original CFTP time with ambiguities. We then prove a general perturbation result, which can be stated informally as follows. Start with an interacting particle system possessing a CFTP time whose definition involves the exploration of an exponentially integrable number of points in the graphical construction, and which satisfies the positive rates property. Then consider a perturbation obtained by adding new transitions to the original dynamics. Our result states that, provided that the perturbation is small enough (in the sense of small enough rates), the perturbed interacting particle system too possesses a CFTP time (with nice properties such as an exponentially decaying tail). The proof consists in defining a CFTP time with ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed dynamics. Finally, we discuss examples of particle systems to which this result can be applied. Concrete examples include a class of neighbor-dependent nucleotide substitution model, and variations of the classical voter model, illustrating the ability of our approach to go beyond the case of weakly interacting particle systems.
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