Corrector theory for elliptic equations with long-range correlated random potential

Guillaume Bal Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027 Josselin Garnier Laboratoire de Probabilit´es et Mod`eles Al´eatoires & Laboratoire Jacques-Louis Lions, Universit´e Paris VII, 2 Place Jussieu, 75251 Paris Cedex 5, France Yu Gu Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027 Wenjia Jing Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027

Analysis of PDEs Probability mathscidoc:2206.03003

Asymptotic Analysis, 77, 123-145, 2012.5
We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. We characterize the deterministic and stochastic correctors. With proper rescaling, the mean-zero stochastic corrector converges to a Gaussian random process in probability and weakly in the spatial variables. In addition, for two prototype equations involving the Laplacian and the fractional Laplacian operators, we prove that the limit holds in distribution in some Hilbert spaces. We also determine the size of the deterministic corrector when it is larger than the stochastic corrector. Depending on the correlation structure of the random field and on the singularities of the Green’s function, we show that either the deterministic or the random part of the corrector dominates.
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@inproceedings{guillaume2012corrector,
  title={Corrector theory for elliptic equations with long-range correlated random potential},
  author={Guillaume Bal, Josselin Garnier, Yu Gu, and Wenjia Jing},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622153741999227446},
  booktitle={Asymptotic Analysis},
  volume={77},
  pages={123-145},
  year={2012},
}
Guillaume Bal, Josselin Garnier, Yu Gu, and Wenjia Jing. Corrector theory for elliptic equations with long-range correlated random potential. 2012. Vol. 77. In Asymptotic Analysis. pp.123-145. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220622153741999227446.
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