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.
- Long range jump
- Quenched invariance principle
- Random conductance model
- Stable-like process
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty