TY - JOUR

T1 - Random conductance models with stable-like jumps

T2 - Quenched invariance principle

AU - Chen, Xin

AU - Kumagai, Takashi

AU - Wang, Jian

N1 - Funding Information:
Funding. The research of Xin Chen is supported by the National Natural Science Foundation of China (No. 11501361 and No. 11871338). The research of Takashi Kumagai is supported by JSPS KAKENHI Grant Numbers JP25247007, JP17H01093 and by the Alexander von Humboldt Foundation.
Funding Information:
The research of Jian Wang is supported by the National Natural Science Foundation of China (No. 11831014 and No. 12071076), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ).
Publisher Copyright:
© 2021 Institute of Mathematical Statistics. All rights reserved.

PY - 2021/6

Y1 - 2021/6

N2 - We study the quenched invariance principle for random conductance models with long range jumps on Zd, where the transition probability from x to y is, on average, comparable to |x − y|−(d+α) with α ∈ (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 α-stable Lévy process on Rd. 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 Hölder regularity of caloric functions for nonelliptic α-stable-like processes on graphs. Our method is robust enough to apply not only for Zd but also for more general graphs whose scaling limits are nice metric measure spaces.

AB - We study the quenched invariance principle for random conductance models with long range jumps on Zd, where the transition probability from x to y is, on average, comparable to |x − y|−(d+α) with α ∈ (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 α-stable Lévy process on Rd. 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 Hölder regularity of caloric functions for nonelliptic α-stable-like processes on graphs. Our method is robust enough to apply not only for Zd but also for more general graphs whose scaling limits are nice metric measure spaces.

KW - Long range jump

KW - Quenched invariance principle

KW - Random conductance model

KW - Stable-like process

UR - http://www.scopus.com/inward/record.url?scp=85108981120&partnerID=8YFLogxK

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U2 - 10.1214/20-AAP1616

DO - 10.1214/20-AAP1616

M3 - Article

AN - SCOPUS:85108981120

SN - 1050-5164

VL - 31

SP - 1180

EP - 1231

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 3

ER -