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

考虑由顶点和边组成的晶格。我们给每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动(或粒子)沿着晶格的顶点移动，使得从一个顶点跳到它的一个相邻顶点的概率与连接边上的电导成正比。这种随机环境中的可逆随机行走模型在文献中称为随机电导模型。几十年来，随机环境中的随机行走一直是概率论研究的中心。一个动机来自于在物理、材料科学或生物学中的应用，例如研究通过多孔介质或在复合材料中的传输过程。这种非均质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质性。由于微观结构往往只能用统计方法来刻画，所以在非均匀介质中的这种传输过程自然地被随机环境中的随机游动所模拟。当在宏观长度和时间尺度上研究这种随机游动时，与微观非均质性相比要大得多，人们通常观察到随机不规则微观结构被平均化并产生均匀化效应，因此在均匀环境中有效的宏观行为可以用一个简单得多的随机过程来描述。数学上，现在感兴趣的是在随机介质上的条件下发生这种均匀化效应。这可以根据随机游走的缩放极限结果来表示。在这个项目中，我们的目标是为具有远距离跳跃的随机电导下的随机游动建立这样的标度极限，即随机游动不仅被允许跳到它的邻居之一，而且被允许跳到更远的其他顶点。作为基本顶点集，我们考虑(1)欧几里德格和(2)点过程在欧几里德空间中的实现。我们还研究了描述这种随机游动转移概率的非局部离散算子的相关偏微分方程组。事实上，随机电导模型对PDE分析师和概率学家都很感兴趣，因为用于研究它们的工具借鉴了这两个领域的技术。它还与数学物理有很强的联系。

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.