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Homogenization of nonlocal equations in randomly evolving media. Diffusion approximation

Marina Kleptsyna, Andrey Piatnitski, Alexandre Popier

TL;DR

The paper analyzes homogenization and diffusion-approximation for a nonlocal convolution-type evolution equation with spatially periodic coefficients and temporally random stationary modulation under diffusive scaling $\alpha=2$. Using two-scale expansions with correctors $\chi_1,\chi_2$ (and higher-order counterparts in the non-symmetric case) and a decomposition of the remainder $R^{\varepsilon}=\varepsilon\big(R^{\varepsilon,(1)}+R^{\varepsilon,(2)}\big)$, it proves that the normalized difference $U^{\varepsilon}=\frac{1}{\varepsilon}(u^{\varepsilon}-\hat{u}^{\varepsilon})$ converges in law to a linear SPDE driven by a Brownian motion, with explicit effective operators $\Theta^{\mathrm{eff}}$, $A^{\mathrm{eff}}$, and $H^{\mathrm{eff}}$. In the non-symmetric case, homogenization in moving coordinates yields a stochastic limit unless a deterministic drift term vanishes (beta condition), and the corresponding effective tensors $\Theta^{\mathrm{eff,ns}}$, $A^{\mathrm{eff,ns}}$, and $H^{\mathrm{eff,ns}}$ are characterized. The results extend nonlocal homogenization theory by providing diffusion-approximation limits and explicit limit equations under precise mixing assumptions.

Abstract

The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the convolution kernel has finite moments up to order three. Under proper mixing assumptions, we study the limit behavior of the normalized difference between solutions of the original and the homogenized problems and show that this difference converges to the solution of a linear stochastic partial differential equation.

Homogenization of nonlocal equations in randomly evolving media. Diffusion approximation

TL;DR

The paper analyzes homogenization and diffusion-approximation for a nonlocal convolution-type evolution equation with spatially periodic coefficients and temporally random stationary modulation under diffusive scaling . Using two-scale expansions with correctors (and higher-order counterparts in the non-symmetric case) and a decomposition of the remainder , it proves that the normalized difference converges in law to a linear SPDE driven by a Brownian motion, with explicit effective operators , , and . In the non-symmetric case, homogenization in moving coordinates yields a stochastic limit unless a deterministic drift term vanishes (beta condition), and the corresponding effective tensors , , and are characterized. The results extend nonlocal homogenization theory by providing diffusion-approximation limits and explicit limit equations under precise mixing assumptions.

Abstract

The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the convolution kernel has finite moments up to order three. Under proper mixing assumptions, we study the limit behavior of the normalized difference between solutions of the original and the homogenized problems and show that this difference converges to the solution of a linear stochastic partial differential equation.
Paper Structure (9 sections, 14 theorems, 108 equations)

This paper contains 9 sections, 14 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. The symmetric case
  4. Equation for the remainder $R^\varepsilon$
  5. Convergence of $R^{\varepsilon,(1)}$
  6. Convergence of $R^{\varepsilon,(2)}$
  7. Proof of Theorem \ref{['thm:main_result']}
  8. The non-symmetric case
  9. Appendix

Key Result

Theorem 1

Under conditions H1-- H5 , there exist such that the process $\dfrac{1}{\varepsilon} (u^\varepsilon-u^0) - \chi_1 \left( \dfrac{x}{\varepsilon}, \dfrac{t}{\varepsilon^2} \right) \nabla u^0$ converges in law in $L^2(\mathbb R^d\times(0,T))$ to the unique solution of the following SPDE: where $u^0$ satisfies the equation with the initial condition $\imath$.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Remark 2
  • ...and 6 more