MEAN 1ST-PASSAGE TIMES OF BROWNIAN MOTION AND RELATED PROBLEMS
MEAN 1ST-PASSAGE TIMES OF BROWNIAN MOTION AND RELATED PROBLEMS
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DOI：
10.1098/rspa.1952.0051
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发表时间：
1952-01-01
影响因子：
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通讯作者：
KLEIN, G
中科院分区：
文献类型：
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作者：
KLEIN, G
一般地发展了布朗运动的首次通过时间理论，并证明对于某些特殊的边界（唯一重要的边界），平均首次通过时间可以很简单地推导出来，避免了通常涉及级数的方法。此外，这些公式与已知的在给定区间内平均“位移”的公式类型有密切的解析关系；存在着若干对相互关系。给出了平移布朗运动的一些有数学意义的新公式。然而，广义理论的主要应用在于推导出旋转布朗运动的实验上特别有用的公式。本文简要讨论了外力存在时的特殊情况和平均相互首次通过时间，最后说明了有限的观测时间如何修正自由布朗运动的平均首次通过时间公式。
The theory of first-passage times of Brownian motion is developed in general, and it is shown that for certain special boundaries—the only ones of any importance—mean first-passage times can be derived very simply, avoiding the usual method involving series. Moreover, these formulae have a close analytical relationship to the better-known type of formulae for average 'displacements’ in given intervals; there exist certain pairs of reciprocal relations. Some new formulae, of mathematical interest, for translational Brownian motion are given. The main application of the general theory, however, lies in the derivation of experimentally particularly useful formulae for rotational Brownian motion. Special cases when external forces are present, and mean reciprocal first-passage times are discussed briefly, and finally it is shown how finite times of observation modify the mean first-passage time formulae of free Brownian motion.