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We consider the dynamics of a tagged particle in an infinite particleenvironment moving according to a stochastic gradient dynamics. For singularinteraction potentials this tagged particle dynamics was constructed first in[FG11], using closures of pre-Dirichlet forms which were already proposed in[GP87] and [Osa98]. The environment dynamics and the coupled dynamics of thetagged particle and the environment were constructed separately. Here wecontinue the analysis of these processes: Proving an essential m-dissipativityresult for the generator of the coupled dynamics from [FG11], we show that thisdynamics does not only contain the environment dynamics (as one component), butis, given the latter, the only possible choice for being the coupled process.Moreover, we identify the uniform motion of the environment as the reversedmotion of the tagged particle. (Since the dynamics are constructed asmartingale solutions on configuration space, this is not immediate.)Furthermore, we prove ergodicity of the environment dynamics, whenever theunderlying reference measure is a pure phase of the system. Finally, we showthat these considerations are sufficient to apply [DMFGW89] for proving aninvariance principle for the tagged particle process. We remark that such aninvariance principle was studied before in [GP87] for smooth potentials, andshown by abstract Dirichlet form methods in [Osa98] for singular potentials.Our results apply for a general class of Ruelle measures corresponding topotentials possibly having infinite range, a non-integrable singularity at 0and a nontrivial negative part, and fulfill merely a weak differentiabilitycondition on $\mathbb R^d\setminus\{0\}$.

An Invariance Principle for the Tagged Particle Process in Continuum with Singular Interaction Potential