Invariant distributions and scaling limits for some diffusions in time-varying random environments

• 成果类型: Article
• 署名作者: Offret, Yoann
• 署名单位: University of Neuchatel
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0475-7
• 发表日期: 2014
• 页码: 1-38
• 关键词: one-dimensional diffusion; random-walk; markov-chains; transition-probabilities; large deviations; random flows; Local Time; SPACE; CONVERGENCE; drift

摘要

We consider a family of one-dimensional diffusions, in dynamical Wiener mediums, which are random perturbations of the Ornstein-Uhlenbeck diffusion process. We prove quenched and annealed convergences in distribution and under weigh-ted total variation norms. We find two kind of stationary probability measures, which are either the standard normal distribution or a quasi-invariant measure, depending on the environment, and which is naturally connected to a random dynamical system. We apply these results to the study of a model of time-inhomogeneous Brox's diffusions, which generalizes the diffusion studied by Brox (Ann Probab 14(4):1206-1218, 1986) and those investigated by Gradinaru and Offret (Ann Inst Henri Poincar, Probab Stat, 2011). We point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations.