Homogenization of random walks: degenerate environments and long-range jumps

随机游走的同质化：退化环境和长程跳跃

• 批准号: EP/W022923/1
• 负责人: Sebastian Andres
• 金额: $ 32.36万
• 依托单位: University of Manchester
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW022923%2F1
• 关键词: Homogenization; random; walks; degenerate; environments

Consider a lattice consisting of vertices and edges. To each edge we assign a randomly chosen positive number called conductance. Now consider a random walk (or particle) moving along the vertices of the lattice in such a way that the probability to jump from one vertex to one of its neighbours is proportional to the conductance on the connecting edge. This model of a reversible random walk in a random environment is known in the literature as the random conductance model. Random walks in random environments have been at the centre of the interest in probability theory for several decades. One motivation originates from applications in physics, material science or biology, as for instance the study of transport processes through porous media or in composite materials. A common characteristics of such heterogeneous media is the presence of strong spatial inhomogeneities on microscopic scales. Since the microscopic structure can often be characterised only statistically, such transport processes in a heterogeneous medium are naturally modelled by random walks in random environment. When studying such random walks on macroscopic length and time-scales, which are much larger compared to the microscopic heterogeneities, one typically observes that the random irregular microstructures are averaged out and homogenisation effects arise, so that the effective macroscopic behaviour can be described by a much simpler stochastic process in a homogeneous environment.Mathematically, it is now of interest under which conditions on the random medium such homogenisation effects occur. This can be formulated in terms of scaling limit results for the random walk. In this project we aim to establish such scaling limits for random walks under random conductances with long range jumps, i.e. the random walk is not only allowed to jump to one of its neighbours but also to other vertices further away. As the underlying set of vertices we consider (1) the Euclidean lattice and (2) the realisation of a point process in the Euclidean space. We also aim to study the associated partial differential differential equations (PDE) involving non-local discrete operators describing the transition probabilities of such random walks. In fact, random conductance models are of interest to PDE analysts as well as probabilists because the tools that are used to study them borrow techniques from both fields. There are also strong links to mathematical physics.

考虑一个由顶点和边组成的晶格。我们给每一条边分配一个随机选择的正数，称为电导。现在考虑一个随机游走（或粒子）沿晶格的顶点沿着移动，使得从一个顶点跳到它的一个邻居的概率与连接边的电导成正比。这种在随机环境中的可逆随机行走的模型在文献中被称为随机电导模型。几十年来，随机环境中的随机游动一直是概率论研究的中心。一个动机源于物理学、材料科学或生物学中的应用，例如研究通过多孔介质或复合材料的传输过程。这种非均匀介质的一个共同特征是在微观尺度上存在强烈的空间不均匀性。由于微观结构通常只能统计地表征，因此在非均匀介质中的这种输运过程自然地由随机环境中的随机行走来模拟。当在宏观长度和时间尺度上研究这种随机行走时，与微观不均匀性相比，宏观长度和时间尺度要大得多，人们通常观察到随机不规则微观结构被平均化并且出现均匀化效应，使得有效的宏观行为可以通过均匀环境中的简单得多的随机过程来描述。现在感兴趣的是，在随机介质上在什么条件下发生这种均化效应。这可以用随机游走的标度极限结果来表示。在这个项目中，我们的目标是建立这样的缩放限制随机电导下的随机游动与长程跳跃，即随机游动不仅允许跳到它的邻居之一，但也到其他顶点更远。作为基本的顶点集，我们考虑（1）欧几里得格和（2）在欧几里得空间中实现点过程。我们还旨在研究相关的偏微分方程（PDE），涉及非局部离散算子描述的转移概率的随机游动。事实上，随机电导模型是偏微分方程分析师和概率学家感兴趣的，因为用于研究它们的工具借鉴了这两个领域的技术。与数学物理也有很强的联系。