Quenched Invariance Principle for Random Walks Among Random Conductances with Stable-Like Jumps

Abstract:
Consider random conductances that allow long range jumps. In particular we consider conductances C_xy=w_xy|x-y|^-d-α for distinct x,y ∈ Z^d and 0<α<2, where {w_xy=w_yx: x,y ∈ Z^d} are positive independent random variables with mean 1. We prove that under some moment conditions for w, suitably rescaled Markov chains among the random conductances converge to a rotationally symmetric α-stable process almost surely w.r.t. the randomness of the environments. The proof is a combination of analytic and probabilistic methods based on the recently established de Giorgi-Nash-Moser theory for processes with long range jumps. This is a joint work with Xin Chen (Shanghai) and Jian Wang (Fuzhou).

Biography:
Takashi Kumagai studied at Kyoto University, where he defended his Ph.D. thesis in 1994. After working at Osaka University and Nagoya University, he went back to Kyoto University in 1998. He is now a professor at the Research Institute for Mathematical Sciences (RIMS), Kyoto University.

Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai