A local time curiosity in random environment
type: Item
instance of: scholarly article
title: A local time curiosity in random environment
zbMATH Open document ID: 0932.60054
DOI: 10.1016/S0304-4149(98)00036-2
author: Zhan Shi
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: A random environment is a system of i.i.d. random variables. A random process is in a random environment if it depends on parameters which are at every time instant given as a realization of the corresponding random variable. The most elementary discrete time random process in random environment is Sinai's random walk. Its continuous time counterpart is the so-called Brox's diffusion process, and the relation is similar to that between the usual random walk and the Brownian motion. Therefore, naturally, both the processes have the same limit distributions. And a similar behaviour would be also expected for the maximum local times which are the quantities defined for a discrete time process \(\{S_k\}_{k\geq 0}\) as \(L^*(n) = \sup_{x\in Z} \sum ^n_{k=0} 1_{\{S_k=x\}}.\) But, surprisingly, the asymptotic behaviour of the discrete and continuous time processes differ substantially which is the main result of the paper.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 1364495
zbMATH Keywords: local time
zbMATH Keywords: Sinai's random walk in random environment
zbMATH Keywords: Brox's diffusion with Brownian potential
reviewed by: Martin Janžura
MaRDI profile type: Publication
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full work available at URL: s0304-4149(98)00036-2
OpenAlex ID: W2015026671
description: scientific article; zbMATH DE number 1364495