Brownian motion on compact groups in infinite dimension
Brownian motion on compact groups in infinite dimension
复制标题
无限维紧群上的布朗运动
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
L. Saloff‐Coste
中科院分区：
文献类型：
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作者：
A. Bendikov;L. Saloff‐Coste
本文讨论紧群上的布朗运动。对于一个紧群G，这些是G值随机过程，具有独立的平稳增量，连续路径，以及一些额外的自然属性，如对称性和双不变性。参见Heyer的条约[26]了解背景和历史评论。我们概述了最近的结果，涉及两类问题：（a）存在的布朗运动具有某些规定的性质，如一维边缘连续密度;（B）之间的关系的各种性质的一维边缘。然后，我们说明这些结果，看看一些具体的例子，紧凑，连接，局部连接的群体，不是李群。我们的总体目标是更好地理解紧群上的布朗运动。下面给出的结果解决了六七十年代出现的一些关于这些过程及其相关的谐波层的问题。例如，哪些群带有双不变的Brelot调和层？哪些是带有布朗运动的群，一维边缘具有连续密度？对于具有连续密度μt（·）的一维边值的布朗运动，当t趋于零时μt（e）的可能行为是什么？在其他自然问题中，有一个问题是研究这些过程的路径的规律性。虽然这将不会在后续讨论，它应该是明确的，我们的结果是在这个方向上的重要基石。我们猜想：任何局部连通的紧致群G的拓扑具有可数基，G中的布朗运动有保持器连续路，其保持器指数为α，且α < 1/2.我们将在即将发表的论文中介绍这些方向的结果。我们的研究自然属于一般领域称为概率代数结构。作为一个确定的研究领域，代数结构上的概率可以追溯到五十年代末和六十年代初。两篇参考文献，其中指出了
In this notes we discuss Brownian motions on compact groups. For a compact group G, these are G-valued stochastic processes having independent stationary increments, continuous paths, and some additional natural properties such as symmetry and bi-invariance. See also Heyer’s treaty [26] for background and historical remarks. We outline recent results concerning two types of problems: (a) existence of Brownian motions having certain prescribed properties such as one-dimensional marginals having continuous densities; (b) relations between various properties of the one-dimensional marginals. We then illustrate these results by looking at some specific examples of compact, connected, locally connected groups that are not Lie groups. Our general goal is to develop a better understanding of Brownian motions on compact groups. The results presented below solve a number of questions that emerged in the sixties and seventies concerning these processes and their associated harmonic sheaves. For instance, which are the groups that carry bi-invariant Brelot harmonic sheaves? which are the groups that carry Brownian motions with one dimensional marginals having a continuous density? For Brownian motions with one dimensional marginals having a continuous density μt(·), what are the possible behaviors of μt(e) as t tends to zero? Among other natural questions is the problem of studying the regularity of the paths of such processes. Although this will not be discussed in the sequel, it should be clear that our results are important building blocks in this direction. We conjecture that any connected locally connected compact group G having a countable basis for its topology admits Brownian motion having Holder continuous paths with Holder exponent α for all α < 1/2. We will present results in these directions in a forthcoming paper. Our study naturally belongs to the general area known as Probability on Algebraic Structures. As an identified field of study, Probability on Algebraic Structures can be traced back to the late fifties and early sixties. Two references with pointers to the early papers on the