Mosco convergence of gradient forms with non-convex potentials II

Abstract

This article provides a scaling limit for a family of skew interacting Brownian motions in the context of mesoscopic interface models. Let M,d ∈ N, y1,...,yM ∈ R, f ∈ Cb(R). For N ∈ N we consider a kN-dimensional, skew reflecting distorted Brownian motion (XN,i )i=1,...,kN , t ≥ 0, and investigate its scaling limit for N →∞.The drift includes skew reflections at height levels ˜ yj := N1−d 2 yj with intensities βj/Nd for j = 1,...,M. The corresponding SDE is given by dXN,i t =−ANXN t i dt − 1 2N−d 2−1 f Nd 2−1XN,i where (BN,i t ) t dt M + j=1 t 1−e−βj/Nd 1+e−βj/Nd dl N,i,˜ yj +dBN,i t , t≥0, i = 1,...,kN, are independent Brownian motions, AN ∈ RkN× kN is symmetric positive definite and l N,i,˜ yj t denotes the local time of (XN,i t the weak convergence of the equilibrium laws of uN t = N ◦XN N2t , t ≥0, ) t≥0 at ˜ yj.Weprove for N →∞,choosing suitable injective, linear maps N : RkN →{h|h : Rd ⊃ D → R}, where D is an open domain. 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)N∈N within that class.