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A Compensator Characterization of Point Processes on TopologicalLattices

We resolve the longstanding question of how to define the compensator of a point process on a general partially ordered set in such a way that the compensator exists, is unique, and characterizes the law of the process. We define a family of one-parameter compensators and prove that this family is unique in some sense and characterizes the finite dimensional distributions of a totally ordered point process. This result can then be applied to a general point process since we prove that such a process can be embedded into a totally ordered point process on a larger space. We present some examples, including the partial sum multiparameter process, single line point processes, multiparameter renewal processes, and obtain a new characterization of the two-parameter Poisson process