论文标题
扩展二维狭窄捕获问题的渐近分析
Asymptotic analysis of extended two-dimensional narrow capture problems
论文作者
论文摘要
在本文中，我们通过考虑将拉普拉斯的渐近扩展转化为每个目标的渐近扩展，扩展了有关具有多个小目标（狭窄捕获问题）的二维（2D）扩散搜索过程的最新工作。后者确定到达时间或捕获时间到单个目标中的分布，并以导致该目标捕获的事件集为条件。 2D中强烈局部扰动的一个特征是，匹配的渐近学在$ν= -1/\lnε$而不是$ε$，$ 0 <ε\ ll 1 $中产生一个系列扩展，其中$ε$指定每个目标的大小相对于搜索域的大小。此外，可以非扰动地上所有对数术语总和。我们利用这一事实表明固定$ν$的Laplace变量$ S $中的泰勒扩展如何为获得条件FPT密度的分裂概率和矩的相应渐近扩展提供了一种有效的方法。然后，我们使用渐近分析为经典狭窄捕获问题的两个主要扩展而得出新的结果：随机重置下的最佳搜索策略，以及在多轮搜索和捕获量下的目标资源积累。
In this paper we extend our recent work on two-dimensional (2D) diffusive search-and-capture processes with multiple small targets (narrow capture problems) by considering an asymptotic expansion of the Laplace transformed probability flux into each target. The latter determines the distribution of arrival or capture times into an individual target, conditioned on the set of events that result in capture by that target. A characteristic feature of strongly localized perturbations in 2D is that matched asymptotics generates a series expansion in $ν=-1/\ln ε$ rather than $ε$, $0<ε\ll 1$, where $ε$ specifies the size of each target relative to the size of the search domain. Moreover, it is possible to sum over all logarithmic terms non-perturbatively. We exploit this fact to show how a Taylor expansion in the Laplace variable $s$ for fixed $ν$ provides an efficient method for obtaining corresponding asymptotic expansions of the splitting probabilities and moments of the conditional FPT densities. We then use our asymptotic analysis to derive new results for two major extensions of the classical narrow capture problem: optimal search strategies under stochastic resetting, and the accumulation of target resources under multiple rounds of search-and-capture.
