The hydrodynamic limit theory of Guo, Papanicolaou and Varadhan suggests a concrete way of analyzing the large-scale behavior of a non-equilibrium interacting particle system. Although the hydrodynamic limit theory has been successfully applied for numerous models, the particle system can be better understood by studying the so-called empirical process. Accordingly, Quastel, Rezakhanlou and Varadhan suggested a way to achieve this by using the symmetric simple exclusion process (SSEP) as a sample model. Despite their methodology being very robust, such an analysis is difficult because of several technical obstacles. Consequently, results were only achieved for two systems: the SSEP and zero-range process (ZRP). Recently, we obtained a third result of this type in the system of locally interacting Brownian motion. This model is a kind of continuum system, whereas the two previous models are lattice systems. Our work verifies that the results of SSEP and ZRP are valid for our model as well.
  We start by introducing the standard methods and classic results of the hydrodynamic limit theory and present our results thereafter.