具有马尔可夫cookie栈的受激随机游动
Excited random walks with Markovian cookie stacks
摘要：
我们考虑$\mathbb{Z}$上的最近邻随机游动，其从站点$x$向右跳转的概率$ω_x(j)$不仅依赖于$x$，而且依赖于先前访问$j$到$x$的次数。集合$(ω_x(j))_{x\in\mathbb{Z},n\ge 0}$有时被称为“cookie环境”，这是由于以下非正式的解释。每次访问一个站点时，步行者从该站点的cookie堆栈中吃一个cookie，并根据所吃cookie的“强度”选择转移概率。我们假设饼干堆是有身份的。并且在站点$x$的栈$(ω_x(j))_{j\ge 0}$内的cookie“强度”遵循有限状态马尔可夫链。因此，每个地点的环境是动态的，但它根据每个地点的步行的本地时间而不是原始的随机步行时间来演变。该模型允许两种不同的制度，关键或非关键，这取决于是否期望的概率跳到右边（或左边）的不变的措施下的马尔可夫链是等于$1/2$或没有。我们表明，在非临界制度的步行总是瞬态的，有非零的线速度，并满足经典的中心极限定理。批判性制度允许更多样化的行为。我们给出了临界状态下行走的重现性/瞬时性和弹道性的充要条件，以及瞬时情况下平均测度下极限定律的完整表征。在本文中考虑的设置推广了以前研究的模型与周期性cookie堆栈。我们的结果的弹道性和极限定理是新的，即使是周期模型。
英文摘要：
We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $ω_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(ω_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the "strength" of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie "strengths" within the stack $(ω_x(j))_{j\ge 0}$ at site $x$ follow a finite state Markov chain. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the original random walk time. The model admits two different regimes, critical or non-critical, depending on whether the expected probability to jump to the right (or left) under the invariant measure for the Markov chain is equal to $1/2$ or not. We show that in the non-critical regime the walk is always transient, has non-zero linear speed, and satisfies the classical central limit theorem. The critical regime allows for a much more diverse behavior. We give necessary and sufficient conditions for recurrence/transience and ballisticity of the walk in the critical regime as well as a complete characterization of limit laws under the averaged measure in the transient case. The setting considered in this paper generalizes the previously studied model with periodic cookie stacks. Our results on ballisticity and limit theorems are new even for the periodic model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
