具有几何迁出的分支过程与“既要有饼干又要吃掉它”随机游走
A Branching Process with Geometric Emigration and the "Have Your Cookie and Eat It" Random Walk
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中文总结 AI 辅助
本文通过引入具有几何迁出的分支过程，研究了一类偏向随机游走的速度与逃逸概率，证实了Pinsky猜想，并计算了相关自交互随机游走的速度与逃逸概率。
中文摘要 AI 辅助
我们考虑整数上的一个随机游走，该游走在每个位置偏向于向右步进，直到首次从该位置向左步进后，此后从该位置向左或向右步进的概率相等，该模型由Pinsky引入。Pinsky计算了该游走的速度以及对于某些p的取值范围该游走逃逸到无穷的概率，并猜想这些公式在更广的范围内成立。为研究该游走，我们引入一个具有几何迁出的分支过程，并回答关于其生命期、暂态与常返性以及当正常返时其平稳分布均值的一些问题。我们将这些结果应用于确认Pinsky的猜想。作为另一个应用，我们计算了一个相关的自交互随机游走的速度和逃逸概率。
英文摘要
We consider a random walk on the integers that at each site is biased to step right until the first time it has stepped left from that site, thereafter stepping left or right from that site with equal chance, introduced by Pinsky. Pinsky calculated the speed of the walk and the probability that such a walk escapes to infinity for some ranges of \(p\) and conjectured these formulas hold over a wider range. To study this walk, we introduce a branching process featuring geometric emigration and answer some questions concerning its life-periods, transience and recurrence, and the mean of its stationary distribution when positive recurrent. We apply these results to confirm Pinsky's predictions. For another application, we calculate the speed and escape probability of a related self-interacting random walk.
发表机构
- Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。
