ON SMALL RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS
ON SMALL RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS
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DOI：
10.1070/rm1970v025n01abeh001254
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发表时间：
1970-02
影响因子：
0.9
通讯作者：
A. Ventsel;M. Freidlin
中科院分区：
文献类型：
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作者：
A. Ventsel;M. Freidlin
在本文中，我们研究了白噪声类型的小随机扰动对动态系统的影响：其中 是维纳过程，as 。我们主要关注这些扰动对随着 的减小而增加的长时间间隔的影响。我们讨论两个问题：第一个是过程不变测度的行为，第二个是轨迹在第一次退出紧致域时的位置分布。在这些问题中，通过估计轨迹不偏离平滑函数超过该时间的概率来发挥重要作用。事实证明，该概率的主项为 小 且具有以下形式，其中 是 的某个非负泛函。函数 是连接 和 的所有函数集合中的最小值，它涉及两个问题的答案。通过引入相空间中独立于扰动的等价关系。我们表明，在某些假设下，不变测度集中在相空间的哪一点上。在这两个问题中，我们通过某个马尔可夫链来近似所讨论的过程；答案取决于与该链关联的图的行为。让我们注意到，第二个问题与最高导数处具有小参数的狄利克雷问题的解的行为密切相关。
In this paper we study the effect on a dynamical system of small random perturbations of the type of white noise: where is the -dimensional Wiener process and as . We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing . We discuss two problems: the first is the behaviour of the invariant measure of the process as , and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of not to deviate from a smooth function by more than during the time . It turns out that the main term of this probability for small and has the form , where is a certain non-negative functional of . A function , the minimum of over the set of all functions connecting and , is involved in the answers to both the problems. By means of we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit. In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of on graphs that are associated with this chain. Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives.