Uniqueness of Dirichlet forms for random point fields in the absence of tail triviality

We consider an infinite system of interacting Brownian motions that preserves a given random point field invariant. Such dynamics are constructed using Dirichlet form theory, which naturally leads to two Dirichlet forms for the random point field: the upper and the lower Dirichlet forms. A fundamental question is the uniqueness of the Dirichlet form: that is, whether these two forms coincide. This uniqueness has often been imposed as a key assumption in the Dirichlet form approach to the stochastic analysis for infinite particle systems. A sufficient condition for the uniqueness of the Dirichlet forms is known when the random point field is tail trivial. However, tail triviality has been established for only a limited class of random point fields. In this paper, we prove the uniqueness of the Dirichlet form without assuming tail triviality. The main contribution of this work is to establish the tail preserving property, which asserts that global properties of the system, such as particle density, are preserved under time evolution. As a consequence, our results also imply the strong uniqueness of solutions to the associated infinite-dimensional stochastic differential equations.