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缘组成的晶格。在每个边缘，我们分配了一个随机选择的正数，称为电导。现在考虑一个随机行走（或粒子）沿晶格的顶点移动，以使从一个顶点跳到其一个邻居之一的概率与连接边缘上的电导率成正比。在文献中，这种可逆随机行走的模型被称为随机电导模型。几十年来，在随机环境中随机步行一直是概率理论兴趣的中心。一种动机源于物理，材料科学或生物学中的应用，例如通过多孔介质或复合材料研究运输过程。这种异质培养基的一个共同特征是微观尺度上存在强空间不均匀性。由于显微镜结构通常只能从统计上进行表征，因此在随机环境中随机步行自然地模拟了这种异质介质中的运输过程。当研究此类随机步行在宏观长度和时间尺度上时，与微观异质性相比，这些随机步行大得多，通常会观察到将随机的不规则微观结构平均散布并出现均匀效应，因此可以通过一个有效的巨镜行为来描述在同质环境中，随机过程中的随机过程要简单得多，从而在随机培养基上发生这种均匀化效果的情况下是感兴趣的。这可以根据随机步行的缩放限量结果来表达。在这个项目中，我们旨在在远距离跳跃的随机电导下建立随机步行的缩放限制，即，随机步行不仅可以跳到其一个邻居之一，而且可以跳到其他顶点。作为基础顶点，我们考虑（1）欧几里得晶格和（2）在欧几里得空间中实现一个点过程。我们还旨在研究涉及描述此类随机步行的过渡概率的非本地离散操作员的相关偏微分方程（PDE）。实际上，PDE分析师和概率的随机电导模型都很感兴趣，因为用于研究它们的工具从这两个领域都借用了技术。与数学物理学有很强的联系。