Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media

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Abstract

Consider diffusions on ℝk, 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 ≪ a2/3. Under a geometric condition on the velocity field β, the final Gaussian phase occurs for times t ≫ a2(log a)2, and the dispersion grows quadratically with a. Under a complementary condition, the final phase shows up at times t ≫ a4(log a)2, or t ≫ a2 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 languageEnglish (US)
Pages (from-to)951-1020
Number of pages70
JournalAnnals of Applied Probability
Volume9
Issue number4
DOIs
StatePublished - Nov 1999

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

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