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 in Dolgopyat and Kosygina (Electron Commun Probab 17:1–14, 2012) (ERWs with boundedly many cookies per stack) and Kosygina and Peterson (Electron J Probab 21:1–24, 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.

我们考虑在无边界递归状态下具有iid马尔可夫饼干堆栈的一维激发随机游走（ERW）。我们证明，在扩散尺度下，这种ERW在标准Skorokhod拓扑中收敛到在其极值（BMPE）处受到扰动的布朗运动的倍数。限制过程的所有参数都是根据单个站点上的Cookie马尔可夫链明确给出的。虽然我们的结果扩展了Dolgopyat和Kosygina（Electron Commun Probab 17：1–14，2012）（每个堆栈中有很多cookie的ERW）以及Kosygina和Peterson（Electron J Probab 21：1–24，2016）（ERW与定期Cookie堆栈），所采用的方法非常不同，并且涉及ERW以及通过漫游更改的随机环境的粗粒度。通过仔细分析每个“介观”步骤后行人留下的环境，我们能够构造出一个“介观”规模的ERW与极限BMPE适当离散的耦合。该分析基于针对ERW的有向边沿局部时间的广义Ray–Knight定理，该局部时间在某些停止时间处停止，并且在原始随机Cookie环境中以及在每次漫游后创建的环境中（这具有更大的挑战性）发展“中观”步骤。