## Abstract

Consider diffusions on ℝ^{k}, k > 1, governed by the Itô equation d X (t)={b(X(t)) + β(X(t)/a)}dt + σdB(t), where b, β are periodic with the same period and are divergence free, σ is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times 1 ≪ t ≪ a^{2/3}. Under a geometric condition on the velocity field β, the final Gaussian phase occurs for times t ≫ a^{2}(log a)^{2}, and the dispersion grows quadratically with a. Under a complementary condition, the final phase shows up at times t ≫ a^{4}(log a)^{2}, or t ≫ a^{2} log a under additional conditions, with no unbounded growth in dispersion as a function of scale. Examples show the existence of non-Gaussian intermediate phases. These probabilisitic results are applied to analyze a multiscale Fokker-Planck equation governing solute transport in periodic porous media. In case b, β are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.

Original language | English (US) |
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Pages (from-to) | 951-1020 |

Number of pages | 70 |

Journal | Annals of Applied Probability |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1999 |

Externally published | Yes |

## Keywords

- Diffusion on a big torus
- Growth in dispersion
- Initial and final Gaussian phases
- Speed of convergence to equilibrium

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty