Institute for Mathematical Physics Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities
Institute for Mathematical Physics Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities Boundaries and Harmonic Functions for Random Walks with Random Transition Probabilities
复制标题
数学物理研究所 具有随机转移概率的随机游走的边界和调和函数 具有随机转移概率的随机游走的边界和调和函数
DOI：
--
复制
发表时间：
--
期刊：
影响因子：
--
通讯作者：
A. S
中科院分区：
文献类型：
--
作者：
V. Kaimanovich;Y. Kifer;Zion Rubshtein;Ben;Typeset By;A. S
组上的通常随机游走（时间和空间上均同质）是由组上的概率度量确定的。在具有随机转移概率的随机游走中，该单个测量被替换为测量的平稳序列，以便生成的（随机）马尔可夫链仍然是空间同质的，但不再是时间同质的。我们研究与该模型相关的各种测量理论边界概念，建立（随机）有界调和函数的泊松公式的模拟，并识别几类群的这些边界。在过去 40 年中，对群体随机游走进行了深入研究（例如，参见 [Ka96] 及其中的参考文献）。它们的重要性归因于众多应用，特别是调和函数的边界和空间的描述以及群作用的遍历性质的研究。这种随机游走是马尔可夫链，它在时间和空间上都是同质的，也可以表示为独立同分布（i.i.d.）群元素的乘积。在随机矩阵乘积的特别重要的情况下，可以使用附加工具，例如李雅普诺夫指数。群 G 上的随机游走由马尔可夫算子 P = P (μ) 确定，该算子包含在 G 上具有固定概率测度 µ 的（右）卷积，因此算子 P 对于左平移对群自身的作用是不变的。有两种模型可以进一步“随机化”这些“普通”随机游走。第一个模型通常称为随机环境中的随机游走 (RWRE)，包括考虑 G 上所有马尔可夫算子空间上的概率测度 λ。在该模型中，各个算子（环境）不是群不变的，尽管通过要求测度 λ 对于 G 在环境空间上的作用准不变来考虑群结构（更具体地说，通常假设 λ 相对于“移动环境”是平移不变的或静止的）链，例如，参见 [Kal81]、[KMo84]、[KSi00]）。根据分布 λ 选择一个随机环境，然后在该环境中运行时间（但不是空间！）齐次马尔可夫链。我们称之为具有随机转移概率的随机游走 (RWRTP) 的另一种模型与 RWRE 相反，因为这里......
The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures , so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model, establish an analogue of the Poisson formula for (random) bounded harmonic functions, and identify these boundaries for several classes of groups. Random walks on groups were intensively studied during the last 40 years (see, for instance, [Ka96] and the references therein). Their importance is due to numerous applications, in particular, to the description of boundaries and spaces of harmonic functions and to the study of ergodic properties of group actions. Such random walks are Markov chains which are homogeneous both in time and space and can be also represented as products of independent identically distributed (i.i.d.) group elements. In the particular important case of products of random matrices additional tools such as Lyapunov exponents can be employed. A random walk on a group G is determined by a Markov operator P = P (µ) which consists in the (right) convolution with a fixed probability measure µ on G, so that the operator P is invariant with respect to the action of the group on itself by left translations. There are two models for further " randomization " of these " ordinary " random walks. The first model is usually referred to as random walks in random environment (RWRE) and consists in considering a probability measure λ on the space of all Markov operators on G. In this model the individual operators (environments) are not group invariant, although the group structure is taken into account by requiring the measure λ to be quasi-invariant with respect to the action of G on the space of environments (more specifically, λ is usually assumed to be either translation invariant or stationary with respect to the " moving environment " chain, see, for instance, [Kal81], [KMo84], [KSi00]). One chooses a random environment according to the distribution λ, and then runs a time (but not space!) homogeneous Markov chain in this environment. The other model which we call random walks with random transition probabilities (RWRTP) is opposite to RWRE in the sense that here one …