A hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as “traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps.

中文翻译：

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具有小陷阱簇的有限或空间周期性二维域中平均首次通过时间的渐近分析

开发了一种混合渐近数值方法来近似平均首次通过时间 (MFPT) 和布朗粒子在有界二维 (2D) 域中的分裂概率，该域包含渐近的吸收盘，称为“陷阱”小半径。与以前要求陷阱在空间上很好分离的研究相比，我们展示了如何通过采用最近制定的最小二乘法来轻松地结合一组紧密间隔的陷阱的影响，以便在数值上解决拉普拉斯附近的某些局部问题集群。我们还为二维空间周期域中的 MFPT 提供了新的渐近公式，其中陷阱簇以斜布拉维格的格点为中心。在所有这些具有固定面积的格子上，对于原始单元，以及每个特定的陷阱集，