Anomalous diffusion via self-interaction and reflection

通过自相互作用和反射的异常扩散

基本信息

批准号：
EP/W006227/1
负责人：
Aleksandar Mijatovic
金额：
$ 58.27万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW006227%2F1
关键词：
Anomalous diffusion via self interaction

项目摘要

Broadly interpreted, a diffusion is a continuous-time stochastic process with continuous trajectories, whose stochastic evolution is driven by a drift and a diffusion coefficient. Diffusions, and their discrete-time cousins, random walks, are ubiquitous in stochastic modelling (e.g., conformation of polymer molecules and roaming of animals) as well as in stochastic optimization algorithms of computational statistics and machine learning. Moreover, diffusions and random walks are prototypical stochastic systems, exhibiting phase transitions in their behaviour depending on values of the underlying parameters of the model. The most well-studied diffusions and random walks have two simplifying features: (i) they are Markov, meaning that the future evolution of the process depends only on the current state, and not its previous history, and (ii) the evolution is homogeneous in space. The proposed research programme will extend the state-of-the-art to non-Markovian and reflecting processes exhibiting anomalous diffusion, in which processes explore space more rapidly than the classical case under assumptions (i) and (ii). A rich and deep classical theory of diffusions and random walks is available. For example, a cornerstone is the result that, in the continuum scaling limit, homogeneous random walks converge to Brownian motion, a universal mathematical object of central importance. This theory extends, in a highly non-trivial way, to broader classes of walks satisfying (i) but a regularity condition weaker than (ii), as shown in recent work of the research team, in which the limit object, Brownian motion, is replaced by a certain class of spatially non-homogeneous diffusion process. While not Brownian motion, these diffusions were nevertheless diffusive, meaning that the rate at which they explore space is the same as in the classical case. Applications motivate more complex models. In one direction, the evolution may depend on the entire history of the walk: to access fresh resources, roaming animals do not retrace their steps, while the excluded volume effect in polymers ensures that no two monomers can occupy the same physical space. In another direction, certain processes are constrained by natural boundaries at which they are forced to reflect or otherwise deviate from the bulk behaviour: queue-length processes in operations research are usually constrained to be non-negative, for example. Both self-interactions and reflections lead to considerably more challenging mathematical models.This proposal sets out an ambitious project to develop novel robust probabilistic techniques for the analysis of certain multidimensional processes exhibiting non-Markovian and/or reflecting behaviour and possessing universal features. The analysis is facilitated by a common underlying structure of these seemingly disparate models, which we identify and then exploit. This structure is easier to identify in the context of continuum models, so that is our focus in this proposal. Once the structure has been identified, we expect these ideas to pave the way for further developments also in discrete versions of the models.

广泛解释，扩散是一个连续的时间随机过程，具有连续轨迹，其随机演化是由漂移和扩散系数驱动的。扩散及其离散的表兄弟，随机步行，无处不在，在随机建模（例如，聚合物分子的构象和动物漫游的构象）以及计算统计和机器学习的随机优化算法中无处不在。此外，扩散和随机步行是原型随机系统，根据模型的基础参数的值，其行为表现出相变。最深思熟虑的扩散和随机步行具有两个简化的特征：（i）它们是马尔可夫，这意味着该过程的未来演变仅取决于当前状态，而不是其先前的历史，并且（ii）（ii）该进化在太空中是均匀的。拟议的研究计划将将最新的最新方法扩展到非马克维亚语，并反映出异常扩散的过程，在该过程中，该过程比假设（i）和（ii）下的经典案例更快地探索了空间。可以提供丰富而深厚的经典理论，即扩散和随机步行。例如，基石是在连续缩放限制的结果中，均匀的随机步行会融合到布朗尼运动，布朗尼运动是一种核心重要性的通用数学对象。该理论以一种高度不平凡的方式扩展到更广泛的步行阶层（i），但正常性条件比（ii）弱，如研究小组的最新工作所示，其中极限对象（Brownian Motion）被一定在空间上非同质性扩散过程所取代。尽管不是布朗运动，但这些扩散是扩散的，这意味着它们探索空间的速率与经典情况相同。应用激发了更复杂的模型。在一个方向上，进化可能取决于步行的整个历史：要获得新的资源，漫游动物不会追溯其步骤，而聚合物中排除的体积效应可确保没有两个单体可以占据相同的物理空间。在另一个方向上，某些过程受到被迫反映或以其他方式偏离批量行为的自然边界的约束：例如，操作研究中的排队长度过程通常被限制为非负值。自我互动和反思都导致了更具挑战性的数学模型。该提案阐明了一个雄心勃勃的项目，旨在开发新颖的强大概率技术，以分析某些具有非马克维亚语和/或反映行为和具有普遍特征的多维过程。这些看似不同的模型的共同基础结构促进了该分析，我们确定并利用这些模型。在连续模型的背景下，这种结构更容易识别，因此这是我们在此提案上的重点。一旦确定了结构，我们希望这些想法在模型的离散版本中也为进一步的发展铺平道路。