This article provides a scaling limit for a family of skew interacting Brownian motions in the context of mesoscopic interface models. Let \(M,d\in \mathbb {N}\) , \(y_1,\dots ,y_M\in \mathbb {R}\) , \(f\in C_b(\mathbb {R})\) . For \(N\in \mathbb {N}\) we consider a \(k_N\) -dimensional, skew reflecting distorted Brownian motion \((X^{N,i}_t)_{i=1,\dots ,k_N}\) , \(t\ge 0\) , and investigate its scaling limit for \(N\rightarrow \infty \) . The drift includes skew reflections at height levels \(\tilde{y}_j:=N^{1-\frac{d}{2}}y_j\) with intensities \(\beta _j/N^d\) for \(j=1,\dots ,M\) . The corresponding SDE is given by \(\begin{aligned}&\text {d} X^{N,i}_t=-\big (A_N X^{N}_t\big )_i\,\text {d} t-\frac{1}{2}N^{-\tfrac{d}{2}-1}\,f\big (N^{\frac{d}{2}-1}X^{N,i}_t\big )\,\text {d} t \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad +\sum _{j=1}^M\tfrac{1-e^{-\beta _j/N^d}}{1+e^{-\beta _j/N^d}}\,\text {d} l_t^{N,i, \tilde{y}_j} +\text {d} B_t^{N,i}, \end{aligned}\) where \({(B_t^{N,i})}_{t\ge 0}\) , \(i=1,\dots , k_N\) , are independent Brownian motions, \(A_N\in \mathbb {R}^{k_N\times k_N}\) is symmetric positive definite and \( l_t^{N,i, \tilde{y}_j}\) denotes the local time of \({(X^{N,i}_t)}_{t\ge 0}\) at \(\tilde{y}_j\) . We prove the weak convergence of the equilibrium laws of \(\begin{aligned} u_{t}^{N}=\Lambda _{N}\circ X^{N}_{N^2t},\quad t\ge 0, \end{aligned}\) for \(N\rightarrow \infty \) , choosing suitable injective, linear maps \(\Lambda _{N}:{\mathbb {R}}^{k_N}\rightarrow \{h\,|\,h:{\mathbb {R}}^d\supset D\rightarrow {\mathbb {R}}\}\) , where D is an open domain. The scaling limit is a distorted Ornstein–Uhlenbeck process whose state space is the Hilbert space \(H=L^2(D,\text {d} z)\) . We characterize a class of height maps, such that the scaling limit of the dynamic is not influenced by the particular choice of \({(\Lambda _{N})}_{N\in {\mathbb {N}}}\) within that class.
