In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $ \mathbb{R}^d $, $ d \geqslant 3 $. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $ (\Phi, \mathcal{R}) $. The point process $ \Phi $ generating the centres of the holes is either a Poisson point process or the lattice $ \mathbb{Z}^d $; the marks $ \mathcal{R} $ generating the radii are unbounded i.i.d random variables having finite $ (d-2+\beta) $-moment, for $ \beta > 0 $. We study the rate of convergence to the homogenized solution in terms of the parameter $ \beta $. We stress that, for low values of $ \beta $, the balls generating the holes may overlap with overwhelming probability.

中文翻译：

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随机穿孔域中泊松问题同质化的收敛率

在本文中，我们提供了在随机穿孔域 $\mathbb{R}^d $, $ d \geqslant 3 $ 中狄利克雷边界条件下泊松问题的均质化的收敛率。我们假设穿过域的孔是球形的，并且是由重新缩放的标记点过程 $ (\Phi, \mathcal{R}) $ 生成的。生成孔中心的点过程 $ \Phi $ 是泊松点过程或格子 $ \mathbb{Z}^d $; 生成半径的标记 $ \mathcal{R} $ 是具有有限 $ (d-2+\beta) $-moment 的无界 iid 随机变量，对于 $ \beta > 0 $。我们根据参数 $\beta $ 研究收敛于同质化解的速率。我们强调，对于 $\beta $ 的低值，产生孔的球可能以压倒性的概率重叠。