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

• 成果类型: Article
• 署名作者: Kosygina, Elena; Mountford, Thomas; Peterson, Jonathon
• 署名单位: City University of New York (CUNY) System; Baruch College (CUNY); Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Purdue University System; Purdue University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01055-3
• 发表日期: 2022
• 页码: 189-275
• 关键词: excited random-walks; central-limit-theorem; repelling random-walk; bond repulsion; speed; times

摘要

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.