We present an analytical treatment of anomalous diffusion in a three-dimensional comb ( xyz -comb) by using the Green’s function approach. We derive exact analytical solutions for the propagators for an instantaneous point injection and natural boundary conditions. The marginal distributions for all three directions are obtained and the corresponding mean squared displacements are found. The analytical results are confirmed by numerical simulations in the framework of coupled Langevin equations. We also analyze a random search process on the xyz -comb, and analytical results on the first arrival time distribution, search reliability and efficiency are obtained. Results for multiple targets are presented as well. The developed approach can be useful for further studies of the Brownian–Lévy searches in the xyz -comb, and analysis of optimal search strategies on comb-like structures.

中文翻译：

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xyz -comb中的异常扩散和随机搜索：确切结果

我们通过使用格林函数方法对三维梳状结构（xyz -comb）中的异常扩散进行分析处理。对于瞬时点注入和自然边界条件，我们得出了传播器的精确解析解。获得所有三个方向的边际分布，并找到相应的均方根位移。在耦合的朗文方程组的框架内，通过数值模拟证实了分析结果。我们还对xyz -comb上的随机搜索过程进行了分析，并获得了首次到达时间分布，搜索的可靠性和效率的分析结果。还显示了多个目标的结果。这种发达的方法可能有助于进一步研究xyz -comb中的Brownian-Lévy搜索，