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

TL;DR

This paper establishes an invariance principle for a tagged particle in a continuum with singular interactions, extending previous results to more general potentials and proving key properties of the coupled dynamics.

Contribution

It proves essential m-dissipativity of the coupled process generator and characterizes the dynamics uniquely, extending invariance principles to broader classes of singular potentials.

Findings

Proved ergodicity of environment dynamics under pure phase measures.

Identified the reversed motion of the tagged particle as the environment's uniform motion.

Established the invariance principle for the tagged particle process under general singular potentials.

Abstract

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction 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 the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from [FG11], we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply [DMFGW89] for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in [GP87] for smooth potentials, and shown by abstract Dirichlet form methods in [Osa98] for singular potentials. Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on $\mathbb R^d\setminus\{0\}$.