Optimal convergence rates in stochastic homogenization in a balanced random environment
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914180432396288 |
|---|---|
| author | Guo, Xiaoqin Tran, Hung V. |
| author_facet | Guo, Xiaoqin Tran, Hung V. |
| contents | We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice $Z^d$ for $d\geq 2$ and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We first derive a quantitative law of large numbers for the invariant measure, which is nearly optimal. A mixing property of the field of the invariant measure is then achieved. We next obtain rates of convergence for the homogenization of the Dirichlet problem for non-divergence form operators, which are generically optimal for $d\geq 3$ and nearly optimal when $d=2$. Furthermore, we establish the existence, stationarity and uniqueness properties of the corrector problem for all dimensions $d\ge 2$. Afterwards, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks, and as a consequence, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01267 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal convergence rates in stochastic homogenization in a balanced random environment Guo, Xiaoqin Tran, Hung V. Probability Analysis of PDEs 35J15 35J25 35K10 35K20 60G50 60J65 60K37 74Q20 76M50 We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in the integer lattice $Z^d$ for $d\geq 2$ and the corresponding problem of stochastic homogenization of non-divergence form difference operators. We first derive a quantitative law of large numbers for the invariant measure, which is nearly optimal. A mixing property of the field of the invariant measure is then achieved. We next obtain rates of convergence for the homogenization of the Dirichlet problem for non-divergence form operators, which are generically optimal for $d\geq 3$ and nearly optimal when $d=2$. Furthermore, we establish the existence, stationarity and uniqueness properties of the corrector problem for all dimensions $d\ge 2$. Afterwards, we quantify the ergodicity of the environmental process for both the continuous-time and discrete-time random walks, and as a consequence, we get explicit convergence rates for the quenched central limit theorem of the balanced random walk. |
| title | Optimal convergence rates in stochastic homogenization in a balanced random environment |
| topic | Probability Analysis of PDEs 35J15 35J25 35K10 35K20 60G50 60J65 60K37 74Q20 76M50 |
| url | https://arxiv.org/abs/2301.01267 |