Dimension-free local convergence and perturbations for reflected Brownian motions
Dimension-free local convergence and perturbations for reflected Brownian motions
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DOI：
10.1214/22-aap1818
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发表时间：
2020-09
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通讯作者：
Sayantan Banerjee;Brendan Brown
中科院分区：
文献类型：
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作者：
Sayantan Banerjee;Brendan Brown
本文描述和分析了$\mathbb{R}^d_+$中的一类正常返反射布朗运动（RBM），其局部统计量以与维数d$无关的速率收敛到平衡点.在适当的假设下的反射矩阵，漂移和扩散系数，维数无关的拉伸指数收敛速度得到估计收缩之间的同步耦合RBM的底层加权距离。我们还研究了对称阿特拉斯模型作为第一步，在获得维数无关的收敛率RBM不满足上述假设。通过分析一个路径导数过程，并将其与随机环境中的随机游动联系起来，得到了对称Atlas模型的差距过程在适当的平稳性扰动下的多项式收敛速度.
We describe and analyze a class of positive recurrent reflected Brownian motions (RBMs) in $\mathbb{R}^d_+$ for which local statistics converge to equilibrium at a rate independent of the dimension $d$. Under suitable assumptions on the reflection matrix, drift and diffusivity coefficients, dimension-independent stretched exponential convergence rates are obtained by estimating contractions in an underlying weighted distance between synchronously coupled RBMs. We also study the Symmetric Atlas model as a first step in obtaining dimension-independent convergence rates for RBMs not satisfying the above assumptions. By analyzing a pathwise derivative process and connecting it to a random walk in a random environment, we obtain polynomial convergence rates for the gap process of the Symmetric Atlas model started from appropriate perturbations of stationarity.