Limit theorems for random walk in random environment

在该方案中，我们讨论了随机环境中的随机行走的极限行为。具体地说，我们研究了d维欧氏格上的随机游动，它的转移概率是由格顶点索引的平稳随机场。大部分重点将放在身份证上。环境。一般来说，RWRE的行为可能与常规随机游走非常不同。在一维和非椭圆多维环境中，人们可能会看到常规随机行走中不存在的现象，例如亚弹道(即，当行走在某个方向上是瞬时的，但以渐近速度零移动时)和亚扩散(即，当从原点到原点的典型距离以显著小于时间的平方根的尺度增长时)。在许多其他情况下，RWRE是一种猜测，其行为类似于常规的随机游走。然而，证明仅在少数情况下为人所知，尤其是对于可逆环境和简单随机游走的小扰动环境。尽管近年来取得了很大的进展，但有关随机随机游动的许多基本问题仍然是悬而未决的，其中最突出的是随机高维椭圆环境中定向瞬变的0-1定律问题和定向瞬变随机游动的平衡性问题。随机游动是概率论中研究最多的对象之一。它的重要性部分源于这样一个事实，即它是许多现实世界现象的自然模型，如扩散、迁移甚至市场波动，并作为研究其他对象的自然工具，如拉普拉斯方程、热导等。虽然简单随机游走的理论已经很发达，但它的大多数技术都严重依赖于它发生的环境的完美规律性。由于在许多真实世界的模型中，这种规律性并不成立，RWRE实际上比简单的随机游走更好地代表了真实世界的现象。因此，我们的目标是开发新的技术，以促进RWRE的研究。

In this proposal we deal with the limit behavior of RWRE (random walks in random environments). Specifically, we look at random walks on the d-dimensional Euclidean lattice, whose transition probabilities are a stationary random field indexed by the vertices of the lattice. Most emphasis will be put on i.i.d. environments. In general the behavior of RWRE can be very different from that of regular random walk. Both in one-dimension and in non-elliptic multidimensional environments, one may witness phenomena that do not exist in regular random walk, such as sub-ballisticity (i.e. when the walk is transient in a certain direction, but move at asymptotic speed zero) and sub-diffusivity (i.e. when the typical distance from the origin grows at a scale that is significantly less than the square-root of the time). In many other cases the RWRE is conjectures to behave similarly to the regular random walk. However, proofs are known only for few cases, most notably for reversible environments and for environments that are small perturbations of the simple random walk. Despite the rapid progress in recent years, many basic questions regarding RWRE are still open, most notably the question of 0-1 law for directional transience and the question of ballisticity for directionally transient random walk in random high-dimensional elliptic environments.Random walks are one of the most studied objects in probability theory. Its importance stems, in part, from the fact that it is a natural model for many real world phenomena such as diffusion, migration and even market fluctuations, and as a natural tool in the study of other objects such as laplace equations, heat conductance and more. While the theory of simple random walks is well developed, most of its techniques rely heavily on the perfect regularity of the environments in which it takes place. Since in many real-world models such regularity does not hold, RWRE is, in fact, a better representation of real-world phenomena than the simple random walk. Thus our aim is to develop new techniques that will facilitate study of RWRE.