Reflection principles for biased random walks and application to escape time distributions
Reflection principles for biased random walks and application to escape time distributions
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有偏随机游走的反射原理及其逃逸时间分布的应用
DOI：
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发表时间：
1985
期刊：
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通讯作者：
V. Balakrishnan
中科院分区：
文献类型：
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作者：
M. Khantha;V. Balakrishnan
我们提出了在半无限和有限链上存在反射障碍的情况下任意偏置连续时间随机游走（包括马尔可夫过程和非马尔可夫过程）的反射原理。对于存在反射屏障的情况下的偏置行走，该原理（不能从组合学推导出来）与存在吸收屏障的情况下其熟悉的形式完全不同。结果使我们能够获得两端吸收和反射障碍的所有三种组合的有限链上偏置行走的条件概率的拉普拉斯变换的闭合形式解。这些解决方案的一个重要应用是计算各种首次通过时间和逃逸时间分布。我们获得了有限链上有偏随机游走的各种逃逸时间分布的特征函数的精确结果。对于受长尾事件时间分布控制的过程，我们表明，即使存在偏差，从有界区域逃逸的平均时间也会发散，从某种意义上说，这表明在这种“冻结”过程中不存在真正的长程扩散。
We present a reflection principle for an arbitrarybiased continuous time random walk (comprising both Markovian and non-Markovian processes) in the presence of areflecting barrier on semi-infinite and finite chains. For biased walks in the presence of a reflecting barrier this principle (which cannot be derived from combinatorics) is completely different from its familiar form in the presence of an absorbing barrier. The result enables us to obtain closed-form solutions for the Laplace transform of the conditional probability for biased walks on finite chains for all three combinations of absorbing and reflecting barriers at the two ends. An important application of these solutions is the calculation of various first-passage-time and escape-time distributions. We obtain exact results for the characteristic functions of various kinds of escape time distributions for biased random walks on finite chains. For processes governed by a long-tailed event-time distribution we show that the mean time of escape from bounded regions diverges even in the presence of a bias—suggesting, in a sense, the absence of true long-range diffusion in such “frozen” processes.