Anomalous diffusion via self-interaction and reflection

广义地解释，扩散是具有连续轨迹的连续时间随机过程，其随机演化由漂移和扩散系数驱动。在随机建模(如聚合物分子构象和动物漫游)以及计算统计和机器学习的随机优化算法中，扩散以及它们的离散近亲随机游动是普遍存在的。此外，扩散和随机游动是典型的随机系统，它们的行为表现出相变，取决于模型的基本参数的值。研究最充分的扩散和随机游动有两个简化的特征：(I)它们是马尔可夫的，这意味着过程的未来演化只取决于当前状态，而不是它以前的历史，以及(Ii)演化在空间上是齐次的。拟议的研究方案将把最新技术扩展到表现出异常扩散的非马尔科夫过程和反映过程，在这些过程中，在假设(一)和(二)下，过程探索空间的速度比经典情况更快。关于扩散和随机游动的丰富而深刻的经典理论已经存在。例如，一个基石是在连续统标度极限下，齐次随机游动收敛到布朗运动的结果，布朗运动是一个具有中心重要性的普遍数学对象。这一理论以一种非常非平凡的方式扩展到满足(I)但比(II)弱的正则性条件的更广泛的游动类别，如研究小组最近的工作所示，其中极限对象布朗运动被一类空间非齐次扩散过程所代替。虽然不是布朗运动，但这些扩散仍然是扩散的，这意味着它们探索空间的速度与经典情况下相同。应用程序推动了更复杂的模型。在一个方向上，进化可能取决于行走的整个历史：为了获得新的资源，漫游的动物不会回溯它们的足迹，而聚合物中排除的体积效应确保了没有两个单体可以占据相同的物理空间。另一方面，某些过程受到自然边界的约束，在自然边界上它们被迫反映或偏离整体行为：例如，运筹学中的排队长度过程通常被约束为非负的。自我相互作用和反思都导致了更具挑战性的数学模型。这项提议提出了一个雄心勃勃的项目，旨在开发新的稳健的概率技术，用于分析某些表现出非马尔科夫和/或反映行为并具有普遍特征的多维过程。这些看似不同的模型的共同底层结构促进了分析，我们识别并利用了这些底层结构。在连续统模型的背景下，这种结构更容易识别，因此这是我们在本提案中的重点。一旦确定了结构，我们预计这些想法将为进一步开发模型的离散版本铺平道路。

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.