QUANTITATIVE HOMOGENIZATION OF THE PARABOLIC AND ELLIPTIC GREEN'S FUNCTIONS ON PERCOLATION CLUSTERS

• 成果类型: Article
• 署名作者: Dario, Paul; Gu, Chenlin
• 署名单位: Tel Aviv University; Universite PSL; Ecole Normale Superieure (ENS)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1456
• 发表日期: 2021
• 页码: 556-636
• 关键词: quenched invariance-principles; random conductance model; simple random-walk; heat-kernel decay; Stochastic Homogenization; large deviations; harnack inequality; partial sums; degenerate; REGULARITY

摘要

We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension d >= 2 and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 127). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. Math. 71 (2018) 1717-1849), Gaussian bounds on the heat kernel established by Barlow in (Ann. Probab. 32 (2004) 3024-3084) and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a C-0,C-1-largescale regularity theory for caloric functions on the infinite cluster and is of independent interest.