RANDOM CONDUCTANCE MODELS WITH STABLE-LIKE JUMPS: QUENCHED INVARIANCE PRINCIPLE

• 成果类型: Article
• 署名作者: Chen, Xin; Kumagai, Takashi; Wang, Jian
• 署名单位: Shanghai Jiao Tong University; Kyoto University; Fujian Normal University; Fujian Normal University; Fujian Normal University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1616
• 发表日期: 2021
• 页码: 1180-1231
• 关键词: parabolic harnack inequality; simple random-walk; upper-bounds; percolation; clusters

摘要

We study the quenched invariance principle for random conductance models with long range jumps on Z(d), where the transition probability from x to y is, on average, comparable to vertical bar x - y vertical bar(-(d+alpha)) with alpha is an element of (0, 2) but is allowed to be degenerate. Under some moment conditions on the conductance, we prove that the scaling limit of the Markov process is a symmetric alpha-stable Levy process on R-d. The well-known corrector method in homogenization theory does not seem to work in this setting. Instead, we utilize probabilistic potential theory for the corresponding jump processes. Two essential ingredients of our proof are the tightness estimate and the Holder regularity of caloric functions for nonelliptic alpha-stable-like processes on graphs. Our method is robust enough to apply not only for Z(d) but also for more general graphs whose scaling limits are nice metric measure spaces.