An effective criterion and a new example for ballistic diffusions in random environment

• 成果类型: Article
• 署名作者: Goergen, Laurent
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP354
• 发表日期: 2008
• 页码: 1093-1133
• 关键词: random-walks

摘要

In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition (T'), introduced by Sznitman [Ann. Probab. 29 (2001) 723-764] and by Schmitz [Ann. Inst. H. Poincare Probab. Statist. 42 (2006) 683-714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when d >= 2, (V) is equivalent to an effective condition that can be checked by local inspection of the environment. When d = 1, we prove that condition (T) is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when d >= 4 a perturbation of Brownian motion by a random drift of size at most epsilon > 0 whose projection on some direction has expectation bigger than epsilon(2-eta), eta > 0, satisfies condition (V) when E is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincare Probab. Statist. 42 (2006) 683-714], (5.4) therein, related to Kalikow's condition.