论文标题
在伯努利环境中概括的多维兴奋随机步行
Generalized Many-Dimensional Excited Random Walk in Bernoulli Environment
论文作者
论文摘要
我们研究了[Ann中引入的$ \ Mathbb {z}^d $上广泛激发随机步行（GERW）的扩展。概率。 40（5），2012年，[7]]撰写的Menshikov，Popov，Ramírez和Vachkovskaia。我们的扩展包括研究激发取决于随机环境的GERW版本。给定的$ p \在（0,1] $（模型的一个参数）中，每当该过程首次访问站点时，概率$ p $就会在给定方向上获得漂移（可能是单位领域的任何方向）。否则，概率$ 1-p $，它在$ d $中的of the a nore a Contere a consection a contery a contercity a contercity a contercity a contercity a contercity a contercity a contercity a contercity a concesity a contercity。 $ d $ - 与零均值的矢量。 $ P $ -GERW是I.I.D.（例如，对于在Bernoulli I.I.D.环境中进行激动的随机步行而满足的条件），我们还获得了大数字和中心限制定理的法律。
We study an extension of the generalized excited random walk (GERW) on $\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov, Popov, Ramírez and Vachkovskaia. Our extension consists in studying a version of the GERW where excitation depends on a random environment. Given $p \in (0,1]$ (a parameter of the model) whenever the process visits a site for the first time, with probability $p$ it gains a drift in a given direction (could be any direction of the unit sphere). Otherwise, with probability $1-p$, it behaves as a $d$-martingale with zero-mean vector. Whenever the process visits an already-visited site, the process acts again as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in short $p$-GERW. Under the same hypothesis of [7] (bounded jumps, uniform ellipticity), we show that the $p$-GERW is ballistic for all $p\in (0,1]$. Under the stronger assumptions that the increments of the regeneration times associated to the $p$-GERW are i.i.d. (condition which is satisfied, for example, for the excited random walk in a Bernoulli i.i.d. environment), we also obtain a Law of Large Numbers and a Central Limit Theorem.
