Mosco convergence of gradient forms with non-convex potentials II

This article provides a scaling limit for a family of skew interacting Brownian motions in the context of mesoscopic interface models. Let d N, y₁, , yM R and f Cb (R) be fixed. For each N N we consider a kN-dimensional, skew reflecting distorted Brownian motion (X^N, iₜ) ₈=₁, , ₊₍, t 0, and investigate the scaling limits for N. The drift includes skew reflections at height levels yⱼ: =N^1-d{2}yⱼ with intensities ⱼ/Nᵈ for j=1, , M. The corresponding SDE is given by equation d X^N, iₜ=- (AN X^Nₜ) ᵢd t-12N^-d{2-1}\, f (N^d{2-1}X^N, iₜ) d t \\+₉=₁M1-e^-ⱼ/Nᵈ1+e^{-ⱼ/Nᵈ}d lₜ^N, i, yⱼ +d Bₜ^N, i, equation where (Bₜ^{N, i) }ₓ ₀, i=1, , kN, are independent Brownian motions and lₜ^N, i, yⱼ denotes the local time of (X^{N, iₜ) }ₓ ₀ at yⱼ. We prove the weak convergence of the equilibrium laws of equation* uₜN=N X^N₍ℂₓ, t 0, equation* for N, choosing suitable injective, linear maps N: R^kN \h\, |\, h: D R\. The scaling limit is a distorted Ornstein-Uhlenbeck process whose state space is the Hilbert space H=L² (D, dz). We characterize a class of height maps, such that the scaling limit of the dynamic is not influenced by the particular choice of (N) ₍ ₍ within that class.