Nonincrease Everywhere of the Brownian Motion Process
Nonincrease Everywhere of the Brownian Motion Process
复制标题
布朗运动过程处处不增
DOI：
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发表时间：
1961
期刊：
影响因子：
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通讯作者：
S. Kakutani
中科院分区：
文献类型：
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作者：
A. Dvoretzky;S. Kakutani
（线性的，可分离的）布朗运动过程已经比任何其他随机过程研究得更多。它有许多应用，至少自巴舍利耶以来，概率学家一直被它微妙而奇特的性质所艾德。它提供，在N的手中。Jyiener，第一个令人满意地定义了n个连续时间参数的非离散随机过程的例子，正是这一工作的Browninn运动（也称为维纳空间），提出了t，他，现在普遍采用的，A. N.柯尔莫哥洛夫定义随机过程。护城河先进的书籍概率投入一些空间t，他的过程，但更微妙的结果超出了他们的范围。一个值得注意的例外是P.L.&Y [2]，它对这一过程作了非常深刻的研究。然而，虽然证明我们的主要结果可以加快呼吁布朗运动的一些先进的工作，但他更喜欢只使用更简单和更好的过程中已知的属性介绍。布朗运动过程可以描述为一个概率空间，其元素都是定义在整条真实的直线上并在原点消失的连续函数。本文的主要目的是证明一个令人意想不到的结果：至少在一点上增加的fw个离子的集合的概率为零。[一个函数被称为在一个点增加，如果它的值稍微到t，这个点的右（左）不是更小（更大）t，比它在点的值。这一结果的正式说明将在下一节中给出，其意义将在下一节中讨论。第4节将给出一个有趣的启发式论证，尽管它是错误的，并导致错误的结果。的
The (linear, separable) Brownian motion process has been studied more than any other stochastic process. It has many applications and, at least since Bachelier, probabilists have been at(tract,ed by its delicate and curious properties. It furnished, in the hands of N. Jyiener, the first instance of a satisfactorily defined nondiscrete stochastic process n-ith continuous time parameter, and it is this work on Browninn motion (also known as Wiener space) that suggested t,he, now universally adopted, method of A. N. Kolmogorov for defining stochastic processes. Moat advanced books on probability devote some space to t,his process but the more delicate results are beyond their scope. A notable exception is P. L&y [2] which contains a x.ery profound study of t,he process. However, though the proof of our principal result could be expedited by appealing to some advanced work on Brownian motion xe preferred a presentation using only the simpler and better known properties of the process. The Brownian motion process can be described as a probability space whose elements are all continuous functions defined on the whole real line and vanishing at the origin. The principal aim of this paper is to pro-e the, to US rather unexpected, result that the probabilify of the set of fw&ions uAich increase at least at one point is zero. [A function is said to increase at a point if its values slightly to t#he right (left) of this point are not smaller (larger) t,han its value at the point,.] A formal statement of this result, will be given in the nest section and its significance will be discussed in the following one. Section 4 will give an interesting, though wrong and leading to a wrong result, heuristic argument. The