Diffusion and random search in heterogeneous media: theory and applications

对生物细胞或无序和多孔介质中主动和被动示踪剂扩散的各种研究表明，环境的基本结构对粒子运动有很大影响，导致粒子运动受限或局部扩散系数和势能函数的变化。因此，为了分析扩散粒子在这种介质中的运动，需要建立考虑环境结构的模型。这产生了非均质扩散模型，该模型通过空间和/或时间相关的扩散系数来描述环境的异质性，反映了不同形式的无序(例如，猝灭的、高能的)。我们考虑的非均质介质包括含有杂质的材料、无定形介质、分形和随机的非均质结构、拥挤的环境、加权图等。因此，本项目中提出的模型将对完全不同的复杂系统中的扩散进行建模。此外，确定最优搜索策略对于从生物学到机器人、从物理到计算机科学的各个领域都是重要的。特别是，随机搜索策略已经被广泛观察到，用于在资源稀缺的环境中觅食的动物、DNA结合蛋白的反应路径、细胞内运输、灾区的搜索操作等。关于信天翁的Levy飞行和LevySearch的最优性的著名争论引发了在动物运动模式和最优策略领域的大量理论和实验研究活动。最优搜索的最新补充是重置机制，偶尔会将粒子重新设置到其起点(例如蜂巢)，提高计算机算法或觅食动物的搜索过程的性能，并减少扩散粒子到达目标的平均第一次通过时间。胶体粒子扩散和重置的实验实现，需要对非均匀介质中的扩散和随机重置随机搜索过程进行进一步的理论研究，特别是在高维环境中。这些非均匀扩散问题与非均匀对流扩散问题和湍流扩散问题密切相关。它们也被广泛应用于金融数学中，特别是布莱克-斯科尔斯模型或CIR和Heston模型背后的几何布朗运动理论。因此，带重置的非均相湍流扩散的结果在经济科学中也是有意义的。在这样的经济系统中，由于自然灾害、流行病或股市崩盘，可能会出现随机重置。

Various studies of active and passive tracer diffusion, e.g., in biologicalcells or in disordered and porous media showed that the underlying structureof the environment has a strong effect on the particle movement, leading toconstrained particle motion or variations of the local diffusion coefficientand potential energy function. Therefore, the development of models accountingfor the structure of the environment is needed for the analysis of diffusiveparticle motion in such media. This gives rise to heterogeneous diffusionmodels which describe the heterogeneity of the environment by space and/ortime-dependent diffusion coefficients, reflecting different forms of disorder(e.g., quenched, energetic). The heterogeneous media we have in mind includematerials with impurities, amorphous media, fractal and random heterogeneousstructures, crowded environments, weighted graphs, etc. Thus the modelsproposed in this project will be of interest for the modelling of diffusionin quite different complex systems.Moreover, to determine optimal search strategies is central for diverse fields,from biology to robotics, from physics to computer science. In particular,random search strategies have been widely observed, for animals foragingin an environment with scarce resources, reaction pathways of DNA-bindingproteins, intracellular transport, search operations in disaster zones,etc. The famed debate on the Levy flight of albatrosses and optimality of Levysearch have triggered a vast amount of theoretical and experimental researchactivities in the field of animal motility patterns and optimal strategies. Amore recent addition to optimal search is the resetting mechanism, theoccasional reset of the particle back to its starting point (the beehive,for instance), increasing the performance of computer algorithms or searchprocesses of foraging animals, and reducing the mean first-passage time ofthe diffusing particle to a target. Experimental realisations of colloidalparticle diffusion and resetting via holographic optical tweezers, requiresfurther theoretical investigation of diffusion and random search processeswith stochastic resetting in heterogeneous media, particularly in higherdimensional settings. These are key objectives in our project.These problems of heterogeneous diffusion are closely related toinhomogeneous advection-diffusion and turbulent diffusion problems. Theyare also heavily used in financial mathematics, especially the theory ofgeometric Brownian motion underlying the Black-Scholes model, or the CIRand Heston models. Therefore, the results obtained for heterogeneous andturbulent diffusion with resetting will be of interest in economic sciencesas well. The presence of stochastic resetting in such economic systems mayoccur due to natural disasters, epidemics or due to stock market crashes.