Mosco convergence of gradient forms with non-convex potentials II

TL;DR

This paper establishes the scaling limit of a family of skew interacting Brownian motions with non-convex potentials, showing convergence to a distorted Ornstein-Uhlenbeck process in a Hilbert space, relevant for mesoscopic interface models.

Contribution

It provides a rigorous weak convergence result for scaled skew Brownian motions with non-convex potentials, extending understanding of mesoscopic interface dynamics.

Findings

Weak convergence of equilibrium laws to a distorted Ornstein-Uhlenbeck process

Characterization of height maps where the limit is independent of the linear maps used

Identification of the scaling limit as a process on a Hilbert space

Abstract

This article provides a scaling limit for a family of skew interacting Brownian motions in the context of mesoscopic interface models. Let $d\in\mathbb N$, $y_1,\dots,y_M\in\mathbb R$ and $f\in C_b(\mathbb R)$ be fixed. For each $N\in\mathbb N$ we consider a $k_N$-dimensional, skew reflecting distorted Brownian motion $(X^{N,i}_t)_{i=1,\dots,k_N}$, $t\geq 0$, and investigate the scaling limits for $N\to\infty$. 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{equation} d X^{N,i}_t=-\big(A_N X^{N}_t\big)_id t-\frac{1}{2}N^{-\tfrac{d}{2}-1}\,f\big(N^{\frac{d}{2}-1}X^{N,i}_t\big)d t \\+\sum_{j=1}^M\tfrac{1-e^{-\beta_j/N^d}}{1+e^{-\beta_j/N^d}}d l_t^{N,i, \tilde y_j} +d B_t^{N,i}, \end{equation} where ${(B_t^{N,i})}_{t\geq 0}$, $i=1,\dots, k_N$, are independent Brownian motions and $ l_t^{N,i, \tilde y_j}$ denotes the local time of ${(X^{N,i}_t)}_{t\geq 0}$ at $\tilde y_j$. We prove the weak convergence of the equilibrium laws of \begin{equation*} u_t^N=\Lambda_N\circ X^{N}_{N^2t},\quad t\geq 0, \end{equation*} for $N\to\infty$, choosing suitable injective, linear maps $\Lambda_N:\mathbb R^{k_N}\to \{h\,|\,h:D\to\mathbb R\}$. The scaling limit is a distorted Ornstein-Uhlenbeck process whose state space is the Hilbert space $H=L^2(D, dz)$. We characterize a class of height maps, such that the scaling limit of the dynamic is not influenced by the particular choice of ${(\Lambda_N)}_{N\in\mathbb N}$ within that class.