### Abstract

We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform C^{0,1}-type estimate and a Liouville theorem of every finite order.

Original language | English (US) |
---|---|

Pages (from-to) | 1945-2014 |

Number of pages | 70 |

Journal | Analysis and PDE |

Volume | 11 |

Issue number | 8 |

DOIs | |

State | Published - Jan 1 2018 |

### Fingerprint

### Keywords

- Large-scale regularity
- Parabolic equation
- Stochastic homogenization
- Variational methods

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics

### Cite this

*Analysis and PDE*,

*11*(8), 1945-2014. https://doi.org/10.2140/apde.2018.11.1945

**Quantitative stochastic homogenization and regularity theory of parabolic equations.** / Armstrong, Scott; Bordas, Alexandre; Mourrat, Jean Christophe.

Research output: Contribution to journal › Article

*Analysis and PDE*, vol. 11, no. 8, pp. 1945-2014. https://doi.org/10.2140/apde.2018.11.1945

}

TY - JOUR

T1 - Quantitative stochastic homogenization and regularity theory of parabolic equations

AU - Armstrong, Scott

AU - Bordas, Alexandre

AU - Mourrat, Jean Christophe

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform C0,1-type estimate and a Liouville theorem of every finite order.

AB - We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform C0,1-type estimate and a Liouville theorem of every finite order.

KW - Large-scale regularity

KW - Parabolic equation

KW - Stochastic homogenization

KW - Variational methods

UR - http://www.scopus.com/inward/record.url?scp=85048301302&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048301302&partnerID=8YFLogxK

U2 - 10.2140/apde.2018.11.1945

DO - 10.2140/apde.2018.11.1945

M3 - Article

AN - SCOPUS:85048301302

VL - 11

SP - 1945

EP - 2014

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 8

ER -