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# Dynamics of Partial Differential Equations

## Volume 8 (2011)

### Number 4

### Stochastic Σ-convergence and applications

Pages: 261 – 310

DOI: http://dx.doi.org/10.4310/DPDE.2011.v8.n4.a1

#### Authors

#### Abstract

Motivated by the fact that in nature almost all phenomena behave randomly in some scales and deterministically in some other scales, we build up a framework suitable to tackle both deterministic and stochastic homogenization problems simultaneously, and also separately. Our approach, the stochastic Σ-convergence, can be seen either as a multiscale stochastic approach since deterministic homogenization theory can be seen as a special case of stochastic homogenization theory (see Theorem 3), or as a conjunction of the stochastic and deterministic approaches, both taken globally, but also each separately. One of the main applications of our results is the homogenization of a model of rotating fluids.

#### Keywords

dynamical system, homogenization supralgebras, stochastic Σ-convergence, Stokes equations

#### 2010 Mathematics Subject Classification

35B40, 35J25, 35R60, 46J10, 60H25