Quantitative homogenization on time-dependent random conductance models with stable-like jumps

Abstract: We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on $\Z<sup>d$, where the transition probability from $x$ to $y$ is given by $w_{t, x,y}|x-y|<sup>{-d-α}$ with $α\in (0,2)$. In particular, time-dependent random coefficients ${w_{t,x,y}: t\in \R_+, (x,y)\in E}$ are uniformly bounded from above (but may be degenerate), and satisfy the Kolmogorov continuous condition, where $E={(x, y): x \not= y \in \Z<sup>d}$ is the set of all unordered pairs on $\Z<sup>d$. The proofs are based on $L<sup>2$-estimates and energy estimates for solutions to regionalparabolic equations and multi-scale Poincaré inequalities associated with time-dependent symmetric stable-like random walks with random coefficients.