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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.15437 |
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| _version_ | 1866929477757435904 |
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| author | Grothaus, Martin Wittmann, Simon |
| author_facet | Grothaus, Martin Wittmann, Simon |
| contents | 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 $β_j/N^d$ 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^{-β_j/N^d}}{1+e^{-β_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=Λ_N\circ X^{N}_{N^2t},\quad t\geq 0, \end{equation*} for $N\to\infty$, choosing suitable injective, linear maps $Λ_N:\mathbb R^{k_N}\to \{h\,|\,h:D\to\mathbb R\}$. 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 ${(Λ_N)}_{N\in\mathbb N}$ within that class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mosco convergence of gradient forms with non-convex potentials II Grothaus, Martin Wittmann, Simon Probability 60J46, 47D07, 82B31 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 $β_j/N^d$ 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^{-β_j/N^d}}{1+e^{-β_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=Λ_N\circ X^{N}_{N^2t},\quad t\geq 0, \end{equation*} for $N\to\infty$, choosing suitable injective, linear maps $Λ_N:\mathbb R^{k_N}\to \{h\,|\,h:D\to\mathbb R\}$. 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 ${(Λ_N)}_{N\in\mathbb N}$ within that class. |
| title | Mosco convergence of gradient forms with non-convex potentials II |
| topic | Probability 60J46, 47D07, 82B31 |
| url | https://arxiv.org/abs/2408.15437 |