Random conductance models with stable-like jumps: Quenched invariance principle

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