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

Xin Chen, Zhen-Qing Chen, Takashi Kumagai, Jian Wang

TL;DR

This paper studies quantitative stochastic homogenization for time-dependent nonlocal (α-stable-like) random conductance models on $\mathbb Z^d$, allowing degenerate and time-correlated coefficients. The authors develop a regional corrector framework and space-time energy estimates, including time-dependent Poincaré and multi-scale Poincaré inequalities, to obtain explicit $L^2$-convergence rates between the scaled parabolic problem with random conductances and the homogenized limit. The main result, Theorem 1.1, provides rates that separate contributions from the time-average convergence $\pi(k^\alpha T)$ and spatial averaging, with rates depending on $\alpha$, $d$, and the decay of test functions encoded by $\beta$. Notably, the work handles degenerate conductances and does not assume ergodicity in time, extending the quantitative homogenization theory to time-dependent nonlocal operators and clarifying how test-function decay and time-averaging influence convergence. The techniques are applicable to both time-dependent and time-independent (via rescaling) nonlocal diffusions, enriching the understanding of stochastic homogenization for stable-like jumps in random environments.

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 $w_{t, x,y}|x-y|^{-d-α}$ 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 $E=\{(x, y): x \not= y \in \Z^d\}$ 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 Poincaré inequalities associated with time-dependent symmetric stable-like random walks with random coefficients.

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

TL;DR

This paper studies quantitative stochastic homogenization for time-dependent nonlocal (α-stable-like) random conductance models on , allowing degenerate and time-correlated coefficients. The authors develop a regional corrector framework and space-time energy estimates, including time-dependent Poincaré and multi-scale Poincaré inequalities, to obtain explicit -convergence rates between the scaled parabolic problem with random conductances and the homogenized limit. The main result, Theorem 1.1, provides rates that separate contributions from the time-average convergence and spatial averaging, with rates depending on , , and the decay of test functions encoded by . Notably, the work handles degenerate conductances and does not assume ergodicity in time, extending the quantitative homogenization theory to time-dependent nonlocal operators and clarifying how test-function decay and time-averaging influence convergence. The techniques are applicable to both time-dependent and time-independent (via rescaling) nonlocal diffusions, enriching the understanding of stochastic homogenization for stable-like jumps in random environments.

Abstract

We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on , where the transition probability from to is given by with . In particular, time-dependent random coefficients are uniformly bounded from above (but may be degenerate), and satisfy the Kolmogorov continuous condition, where is the set of all unordered pairs on . The proofs are based on -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.

Paper Structure

This paper contains 9 sections, 10 theorems, 183 equations.

Table of Contents

  1. Introduction and main result
  2. Parabolic equations associated with the operator $\mathcal{L}_t^{k,\omega}$
  3. Poincaré inequality and multi-scale Poincaré inequality
  4. Large scale estimates for correctors
  5. Quantitative differences between the scaled operators $\{\mathcal{L}_t^{k,\omega}\}_{k\geqslant1}$ and the limit $\bar{\mathcal{L}}$
  6. Quantitative homogenizations
  7. Appendix
  8. Dense property of $\mathscr{S}_{\infty, g,T}$
  9. Proofs of Poincaré-type inequalities

Key Result

Theorem 1.1

Assume that Assumption (H) holds and that $d>\alpha$. For $T>0$, $g\in C_c^2 (\mathbb R^d)$, $\beta\in(0,\infty]$ and $h\in \mathscr{S}_{\beta, g,T}$, let $\bar{u}$ and $u^\omega_k$ be the solutions to e6-3 and e6-4, respectively. Then, for a.e. $\omega\in \Omega$ and any $\gamma>0$, there are a co where $\pi(t):=\frac{1}{t} \int_0^t \left|\mathcal{K} (s)-\mathcal{K} \right|^2\,ds$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 17 more