Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields. (English) Zbl 1523.35025
Summary: We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale \(R\ge 1\). For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.
MSC:
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
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