The classical Dirichlet form given by the intrinsic gradient on Γℝd is associated with a Markov process consisting of a countable family of interacting diffusions. By considering each diffusion as a particle with unit mass, the randomly evolving configuration can be thought of as a Radon measure valued diffusion. The quasi-sure analysis of Dirichlet forms is used to find exceptional sets of configurations for this Markov process. We consider large scale properties of the configuration and show that, for quite general measures, the process never hits those unusual configurations that violate the law of large numbers. Furthermore, for certain Gibbs measures, which model random particles in ℝd that interact via a potential function, we show, for d=1, 2, that the process never hits those unusual configurations that violate the law of the iterated logarithm.
The Law of the Iterated Logarithm for a Markov Process
