Anomalous stochastic search

背景：生物动力学的特征是活动，即能量从环境中吸收并转化为运动[1]。自行式运动的例子从人类的运动到觅食的海洋捕食者，再到爬行的生物细胞。主动动力学与被动布朗运动非常不同，在被动布朗运动中，示踪粒子是通过与周围流体粒子的碰撞来驱动的。此外，生物过程经常表现出以长期扩散为特征的异常动力学，这再次与布朗运动非常不同[2]。这反映了生物系统的时空复杂性。在数学上，活动和异常动力学都是按照高级(持续、非马尔科夫、非高斯)随机过程来建模的[1，2]。这个项目将交叉这两个最新的新研究领域，以了解对目标的搜索，例如，觅食过程中的食物源[3]。项目描述：热身将学习L走，这是一类著名的异常随机过程的基本类别[4]。它们的扩散特性应该在简单的搜索场景中以解析的方式理解并以数字的方式进行探索，以数学形式表示为首次通过和首次到达问题。虽然人们对一维空间中的这些动力学了解很多，但更高维度的环境处于研究的前沿。这些知识应该被用来理解G.Volpe(伦敦)和V.Trianni(罗马)关于生物和机器人L步行者的新实验数据。应该通过开发和分析新的随机模型来探索生物活动和单个粒子之间的相互作用对搜索效率的影响。参考文献[1]C.Bechinger，R.Di Leonardo，H.Lowen，C.Reichhardt G.Volpe，G.Volpe，Rev.Mo.Phys.88,045006(2016年)[2]R.Klages，G.Radons，I.M.Sokolov(编辑)，异常运输：基础和应用(Wiley-Vch，柏林，2008年)[3]R.Klages，寻找鸟类、鱼类和昆虫的食物，书中的章节：A.Bunde，J.Caro，J.Kerger，G.Vogl(编辑)，Diffusive Prospoping in Natural，Technology and Social。(Springer，柏林，2017)[4]V.Zaburdaev，S.Denisv，J.KlAfter，Rev.Mo.Phys.87,483(2015)

Background: Biological dynamics are characterised by activity in the sense that energy is taken up from the environment and converted into motion [1]. Examples of self-propelled motion range from human movement, to foraging marine predators, to crawling biological cells. Active dynamics are very different from passive Brownian motion, where a tracer particle is driven by collisions with the surrounding fluids particles. Biological processes furthermore often exhibit anomalous dynamics characterised by long-term diffusion that is, again, very different from Brownian motion [2]. This reflects the spatio-temporal complexity of biological systems. Mathematically both active and anomalous dynamics are modeled in terms of advanced (persistent, non-Markovian, non-Gaussian) stochastic processes [1,2]. This project will cross-link these two very recent, new fields of research for understanding the search of targets like, e.g., food sources in a foraging process [3].Project description: The warm-up will be to learn about Lévy walks, a famous fundamental class of anomalous stochastic processes [4]. Their diffusive properties should be understood analytically and explored numerically in simple search scenarios, formulated mathematically as first passage and first arrival problems. While much is known about these dynamics in one dimension, higher dimensional settings are at the forefront of research. This knowledge should be applied to understand new experimental data by G.Volpe (London) and V.Trianni (Rome) on biological and robotic Lévy walkers. The impact of biological activity and interactions between single particles on search efficiency should be explored by developing and analysing new stochastic models. References[1] C.Bechinger, R.Di Leonardo, H.Lowen, C.Reichhardt G.Volpe, G.Volpe, Rev.Mod.Phys. 88, 045006 (2016)[2] R.Klages, G.Radons, I.M.Sokolov (Eds.), Anomalous Transport: Foundations and Applications (Wiley-VCH, Berlin, 2008)[3] R.Klages, Search for food of birds, fish and insects, book chapter in: A.Bunde, J.Caro, J.Kaerger, G.Vogl (Eds.), Diffusive Spreading in Nature, Technology and Society. (Springer, Berlin, 2017)[4] V. Zaburdaev, S. Denisov, J. Klafter, Rev.Mod.Phys. 87, 483 (2015)