论文标题
随机步行与马尔可夫·饼干堆栈的随机步行融合到极限扰动的布朗尼运动
Convergence of random walks with Markovian cookie stacks to Brownian motion perturbed at extrema
论文作者
论文摘要
我们考虑使用I.I.D.的一维激发随机步行（ERW）。马尔可夫·饼干在非边界反复发生的政权中堆叠。我们证明，在扩散缩放下，这样的ERW在标准的Skorokhod拓扑中收敛到其极端（BMPE）扰动的Brownian Motion的倍数。限制过程的所有参数均以单个站点的Cookie Markov链的链接明确给出。尽管我们的结果扩展了Dolgopyat和Kosygina的结果（2012年，每个站点的ERWS有限多）以及Kosygina和Peterson（2016年，带有定期饼干堆栈的ERW），但采用的方法却非常不同，并且涉及ERW和随行的环境的粗粒子，并且随着步行而改变了。通过在每个``介质''步骤之后对步行所留下的环境进行仔细的分析，我们能够在此``介质''量表上构建ERW的耦合，并通过适当的限制BMPE离散。该分析基于对ERW的定向边缘时代的通用射线骑士定理，在某些停止时间停止，并且在原始的随机曲奇环境中都在不断发展，并且（这更具挑战性）在每次``Messoscopic''步骤后创造的环境中都会发展。
We consider one-dimensional excited random walks (ERWs) with i.i.d. markovian cookie stacks in the non-boundary recurrent regime. We prove that under diffusive scaling such an ERW converges in the standard Skorokhod topology to a multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of the limiting process are given explicitly in terms of those of the cookie markov chain at a single site. While our results extend the results of Dolgopyat and Kosygina (2012, ERWs with boundedly many cookies per site) and Kosygina and Peterson (2016, ERWs with periodic cookie stacks), the approach taken is very different and involves coarse graining of both the ERW and the random environment changed by the walk. Through a careful analysis of the environment left by the walk after each ``mesoscopic'' step, we are able to construct a coupling of the ERW at this ``mesoscopic'' scale with a suitable discretization of the limiting BMPE. The analysis is based on generalized Ray-Knight theorems for the directed edge local times of the ERW stopped at certain stopping times and evolving in both the original random cookie environment and (which is much more challenging) in the environment created by the walk after each ``mesoscopic'' step.
