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 (XN,iₜ)i=1,,kN, t≥ 0, and investigate the scaling limits for N→∞. The drift includes skew reflections at height levels yⱼ:=N1-d/2yⱼ with intensities βⱼ/Nᵈ for j=1,,M. The corresponding SDE is given by {equation} d XN,i_t=-(A_N XN_t)_id t-1/2N-d/2-1\,f(Nd/2-1XN,i_t)d t \\+∑ⱼ₌₁^M{1-e-β_j/N^d}{1+e-β_j/N^d}d l_tN,i, y_j +d B_tN,i, {equation} where (BₜN,i)t≥ 0, i=1,, kN, are independent Brownian motions and lₜN,i, yⱼ denotes the local time of (XN,iₜ)t≥ 0 at yⱼ. We prove the weak convergence of the equilibrium laws of {equation*} u_t^N=Λ_N∘ XNN^2t, t≥ 0, {equation*} for N→∞, choosing suitable injective, linear maps ΛN: RkN→ \,|\,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 (ΛN)N∈ N within that class.