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

• 成果类型: Article
• 署名作者: Bhattacharya, R
• 署名单位: Indiana University System; Indiana University Bloomington
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 1999
• 页码: 951-1020
• 关键词: central limit-theorem; gradient tracer test; spatial moments; cape-cod; dispersion; aquifers; scale; sand; Massachusetts; asymptotics

摘要

Consider diffusions on R-k, k > 1, governed by the It (o) over cap equation dX(t) = {b(X(t)) + beta(X(t)/a)} dt + sigma dB(t), where b, beta are periodic with the same period and are divergence free, cr is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times 1 much less than t much less than a(2/3). Under a geometric condition on the velocity field beta, the final Gaussian phase occurs for times t much greater than a(2)(log a)(2), and the dispersion grows quadratically with a. Under a complementary condition, the final phase shows up at times t much greater than a(4)(log a)(2), or t much greater than 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, beta are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.