We construct N -particle Langevin dynamics in R d or in a cuboid region with periodic boundary for a wide class of N -particle potentials and initial distributions which are absolutely continuous w.r.t. Lebesgue measure. The potentials are in particular allowed to have singularities and discontinuous gradients (forces). An important point is to prove an L p -uniqueness of the associated non-symmetric, non-sectorial degenerate elliptic generator. Analyzing the associated functional analytic objects, we also give results on the long-time behaviour of the dynamics, when the invariant measure is finite: Firstly, we prove the weak mixing property whenever it makes sense (i.e. whenever { < ∞} is connected). Secondly, for a still quite large class of potentials we also give a rate of convergence of time averages to equilibrium when starting in the equilibrium distribution. In particular, all results apply to N -particle systems with pair interactions of Lennard-Jones type.

We construct an N-particle Langevin dynamics on a cuboid region in R d with periodic boundary condition, i.e., a diffusion process solving the Langevin equation with periodic boundary condition in the sense of the corresponding martingale problem. Our approach works for general H 1,∞ potentials allowing Nparticle interactions and external forces. Of course, the corresponding forces are not necessarily continuous. Since the generator of the dynamics is non-sectorial, for the construction we use the theory of generalized Dirichlet forms.

We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in R d having velocities as marks. The construction is done via a limiting procedure using N -particle dynamics in cubes (−λ, λ] d with periodic boundary conditions. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N -particle systems in (−λ, λ] d with periodic boundary conditions. After proving tightness of the laws of finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space (and their weak limit) fulfilling a uniform Ruelle bound. Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for general repulsive interaction potentials φ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions d ≥ 1.Date: October 26, 2018.

Keywords :Smooth contractive extension operator, elliptic differential operator, Feynman formula, Chernoff theorem. AbstractWe prove two assumptions made in an article by Ya.A. Butko, M. Grothaus, O.G. Smolyanov concerning the existence of a strongly continuous operator semigroup solving a Cauchy-Dirichlet problem for an elliptic differential operator in a bounded domain and the existence of a smooth contractive embedding of a core of the generator of the semigroup into the space C 2,α c (R n ). Based on these assumptions a Feynman formula for the solution of the Cauchy-Dirichlet problem is constructed in the article mentioned above. In this article we show that the assumptions are fulfilled for domains with C 4,α -smooth boundary and coefficients in C 2,α .