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

TL;DR

This paper proves quantitative homogenization for time-dependent random conductance models with stable-like jumps on integer lattices, providing bounds and estimates for long-range jump processes with random, possibly degenerate coefficients.

Contribution

It introduces new homogenization results for time-dependent stable-like jump processes with random conductances, extending previous work to include degeneracy and time dependence.

Findings

Established quantitative homogenization results for stable-like jumps

Derived $L^2$ and energy estimates for solutions to related equations

Provided multi-scale Poincaré inequalities for time-dependent models

Abstract

We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on $\Z^d$, where the transition probability from $x$ to $y$ is given by $w_{t, x,y}|x-y|^{-d-\alpha}$ with $\alpha\in (0,2)$. 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 $E=\{(x, y): x \not= y \in \Z^d\}$ is the set of all unordered pairs on $\Z^d$. The proofs are based on $L^2$-estimates and energy estimates for solutions to regionalparabolic equations and multi-scale Poincar\'e inequalities associated with time-dependent symmetric stable-like random walks with random coefficients.