Diffusions on an infinite dimensional torus
Diffusions on an infinite dimensional torus
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DOI：
10.1016/0022-1236(81)90047-1
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发表时间：
1981-06
影响因子：
1.7
通讯作者：
R. Holley;D. Stroock
中科院分区：
文献类型：
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作者：
R. Holley;D. Stroock
设T是平面上的一维单位圆。本文研究无限维环TZD上一类特殊的对称扩散过程。我们特别关心的是对这些扩散的平稳分布的分析。除了这些扩散与统计力学中称为平面转子模型的某种连续状态的伊辛型模型之间的联系外，我们认为这种扩散的研究在纯粹的数学基础上是有趣的。事实上，就我们所知，无限维扩散的遍历理论目前还知之甚少。我们在这里理解不足的基本原因是来自统计力学的模型通常给出的一个原因：我们真的不知道如何处理无限维随机动力系统，在这个系统中，每个坐标的活动与在其他坐标中的活动一样强烈。当面对这样的系统时，一个人必须放弃所有为处理有限维情况而开发的强大的机器，因为对于有限维过程的合理假设(例如，Harris条件)在处理即使是最简单的无限维过程时显然也是不合理的。因此，当一个人想要研究这些无限维度的扩散时，他的技术选择就会非常有限。在这篇文章中，我们将利用少数几种已被证明对研究起源于统计力学的其他随机过程有用的技术之一。我们将使用的技术对统计机械师和经典动力学的学者来说都是熟悉的；即，我们将使用统计机械师所知的Liapunov函数作为特殊的Jic能量泛函。在目前的上下文中，Liapunov函数是
Let T be the one dimensional unit circle in the plane. In this paper we are going to study a special class of symmetric diffusion processes on the infinite dimensional torus TZd. Of particular concern to us will be the analysis of the stationary distributions for these diffusions. Aside from the connection which these diffusions have with a certain continuous state Ising-type model in statistical mechanics known as the plane rotor model, we believe that the study of such diffusions is interesting on purely mathematical grounds. Indeed, so far as we know, the ergodic theory of infinite dimensional diffusions is as yet poorly understood. The basic reasons for the poverty of our understanding here is the usual one given for models coming from statistical mechanics: we do not really know how to handle infinite dimensional stochastic dynamical systems in which the activity of each coordinate is just as intense as it is in every other coordinate. When confronting such systems, one must necessarily abandon all the powerful machinery developed to handle the finite dimensional case, because what are reasonable assumptions for finite dimensional processes (eg, the condition of Harris) are patently unreasonable when dealing with even the simplest infinite dimensional processes. Consequently, one’s choice of techniques is very much curtailed when one wants to study these infinite dimensional diffusions. In this paper we will exploit one of the few techniques which has proved itself to be useful in the study of other stochastic processes whose origins are in statistical mechanics.The technique which we will be using is familiar to both statistical mechanicians and afficiandos of classical dynamics; namely, we are going to be using a Liapunov function known to statistical mechanicians as the speciJic energy functional. In the present context, a Liapunov function is a