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We consider one-dimensional excited random walks (ERWs) with i.i.d. markoviancookie stacks in the non-boundary recurrent regime. We prove that underdiffusive scaling such an ERW converges in the standard Skorokhod topology to amultiple of Brownian motion perturbed at its extrema (BMPE). All parameters ofthe limiting process are given explicitly in terms of those of the cookiemarkov chain at a single site. While our results extend the results ofDolgopyat and Kosygina (2012, ERWs with boundedly many cookies per site) andKosygina and Peterson (2016, ERWs with periodic cookie stacks), the approachtaken is very different and involves coarse graining of both the ERW and therandom environment changed by the walk. Through a careful analysis of theenvironment left by the walk after each ``mesoscopic'' step, we are able toconstruct a coupling of the ERW at this ``mesoscopic'' scale with a suitablediscretization of the limiting BMPE. The analysis is based on generalizedRay-Knight theorems for the directed edge local times of the ERW stopped atcertain stopping times and evolving in both the original random cookieenvironment and (which is much more challenging) in the environment created bythe walk after each ``mesoscopic'' step.

Convergence of random walks with Markovian cookie stacks to Brownian motion perturbed at extrema