Construction of N-Particle Langevin Dynamics for H 1,∞-Potentials via Generalized Dirichlet Forms

Abstract

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.

Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials

“…As in [5,Lemma 7] using perturbation theory one concludes that (L ,D), defined byL =L 0 − ∇ ∇ v , is essentially m-dissipative.…”

“…Note that also in this case the forces ∇φ in (1.1) may be non-continuous. The case M = d i=1 [0, r i ] is treated in [5]. For the definition of a Dirichlet operator and related information see e.g.…”

“…To prove the essential m-dissipativity, we proceed similarly as in [5] (therefore this proof is not very detailed). We switch to the complexified setting, so L 2 (E; μ ) and other function spaces consist of complex valued functions.…”

“…In [5] we constructed a Hunt process weakly solving (1.1) for most inital points in the case where M = d i=1 [0, r i ] and ∈ H 1,∞ (M N ) (i.e. is Lipschitz continuous, but ∇ may be non-continuous) using the theory of generalized Dirichlet forms (cf.…”

“…(We are quite sure that (P2) can be weakened at least to continuity of e − on some M N \ (M N ) K , K ∈ R.) We use (P2) for approximating pointwisely a.e. on M N by smoothed versions n of the cutoffs ∧ n, such that the results from [5] and Sect. 2 apply to n and such that n − ∧ n ∞ becomes small as n → ∞ (or is at least bounded uniformly in n ∈ N).…”