The Annals of Applied Probability

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

Rabi Bhattacharya

Full-text: Open access


Consider diffusions on $\mathbb{R}^k > 1$, governed by the Itô equation $dX(t) = {b(X(t)) + \beta(X(t)/a)} dt + \sigmadB(t)$, where $b, \beta$ are periodic with the same period and are divergence free, $\sigma$ is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times $1 \ll t \ll a^{2/3}$. Under a geometric condition on the velocity field $\beta$, the final Gaussian phase occurs for times $t \gg a^2(\log a)^2$, and the dispersion grows quadratically with a . Under a complementary condition, the final phase shows up at times $t \gg a^4(\log a)^2$, or $t \gg a^2 \log a$ 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.

Article information

Ann. Appl. Probab., Volume 9, Number 4 (1999), 951-1020.

First available in Project Euclid: 21 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F60 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Diffusion on a big torus speed of convergence to equilibrium initial and final Gaussian phases growth in dispersion


Bhattacharya, Rabi. Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. Ann. Appl. Probab. 9 (1999), no. 4, 951--1020. doi:10.1214/aoap/1029962863.

Export citation


  • Adams, E. E. and Gelhar, L. W. (1992). Field study of dispersion in a heterogeneous aquifer. 2. Spatial moments analysis. Water Resour. Res. 28 3293-3307.
  • Aronson, D. G. (1967). Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 890-896.
  • Bensoussan, A., Lions, J. L. and Papanicolaou, G. C. (1978). Asy mptotic Analy sis for Periodic Structures. North-Holland, Amsterdam.
  • Bhattachary a, R. N. (1982). On the functional central limit theorem and the law of iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete. 60 185-201.
  • Bhattachary a, R. N. (1985). A central limit theorem for diffusions with periodic coefficients. Ann. Probab. 13 385-396.
  • Bhattachary a, R. N., Denker, M. and Goswami, A. (1999). Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales. Stochastic Process. Appl. 80 55-86.
  • Bhattachary a, R. N. and G ¨otze, F. (1995). Time-scales for Gaussian approximation and its break down under a hierarchy of periodic spatial heterogeneities. Bernoulli 1 81-123.
  • Bhattachary a, R. N. and Gupta, V. K. (1979). On a statistical theory of solute transport in porous media. SIAM J. Appl. Math. 37 485-498.
  • Bhattachary a, R. N. and Gupta, V. K. (1983). A theoretical explanation of solute dispersion in saturated porous media at the Darcy scale. Water Resour. Res. 19 938-944.
  • Bhattachary a, R. N., Gupta, V. K. and Walker, H. F. (1989). Asy mptotics of solute dispersion in periodic porous media. SIAM J. Appl. Math. 49 86-98.
  • Bhattachary a, R. N. and Ramasubramanian, S. (1988). On the central limit theorem for diffusions with almost periodic coefficients. Sankhy¯a Ser. A 50 9-25.
  • Bhattachary a, R. N. and Way mire, E. C. (1990). Stochastic Processes with Applications. Wiley, New York.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Chen, M. F. and Wang, F. Y. (1994). Applications of coupling method to the first eigenvalue on a manifold. Sci. Sin. (A) 37 1-14. (English edition.)
  • Chen, M. F. and Wang, F. Y. (1997). Estimates of the logarithmic Sobolev constant. An improvement of Bakry-Emery criterion. J. Funct. Anal. 144 287-300.
  • Cushman, J. (ed.) (1990). Dy namics of Fluid in Hierarchical Porous Media. Academic Press, New York.
  • Dagan, G. (1984). Solute transport in heterogeneous porous formations. J. Fluid Mech. 145 151- 177.
  • Diaconis, P. and Stroock, D. W. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36-61.
  • Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695-750.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • Fannjiang, A. and Papanicolaou, G. (1994). Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 333-408.
  • Fill, J. A. (1991). Eigenvalue bounds on convergence to stationarity for nonreversible Markov chains, with an application to the exclusion process. Ann. Appl. Probab. 1 62-87.
  • Fried, J. J. and Combarnous, M. A. (1971). Dispersion in porous media. Adv. Hy drosci. 7 169- 282.
  • Friedman, A. (1975). Stochastic Differential Equations and Applications 1. Academic Press, New York.
  • Garabedian, S. P., LeBlanc, D. R., Gelhar, L. W. and Celia, M. A. (1991). Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts 2. Analy sis of spatial moment for a nonreactive tracer. Water Resour. Res. 27 911-924.
  • Gelhar, L. W. and Axness, C. L. (1983). Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19 161-180.
  • Gupta, V. K. and Bhattachary a, R. N. (1986). Solute dispersion in multidimensional periodic saturated porous media. Water Resour. Res. 22 156-164.
  • Guven, O. and Molz. F. J. (1986). Deterministic and stochastic analysis of dispersion in an unbounded stratified porous medium. Water Resour. Res. 22 1565-1574.
  • Hall, P. G. and Hey de, C. C. (1980). Martingale Central Limit Theory and Its Applications. Academic Press, New York.
  • Holley, R. A., Kusuoka, S. and Stroock, D. W. (1989). Asy mptotics of the spectral gap, with applications to simulated annealing. J. Funct. Anal. 83 333-347.
  • Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, New York.
  • Kozlov, S. M. (1979). Averaging operators with almost periodic, rapidly oscillating coefficients. Math. USSR-Sb. 35 481-498.
  • Kozlov, S. M. (1980). Averaging of random operators. Math. USSR-Sb. 37 167-180. LeBlanc, R. D., Garabedian, S. P., Hess, K. M., Gelhar, L. W., Quadri, R. D., Stollenwerk,
  • K. G. and Wood, W. W. (1991). Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts 1. Experimental design and observed tracer movement. Water Resour. Res. 27 895-910.
  • Nagaev, S. V. (1961). More exact statements of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6 62-81.
  • Papanicolaou, G. C. and Varadhan, S. R. S. (1979). Boundary problems with rapidly oscillating coefficients. Colloq. Math. Soc. J´anos Boly ai 27 835-875.
  • Reed, M. and Simon, B. (1980). Methods of Modern Mathematical physics 1. Functional Analy sis, rev. ed. Academic Press, New York.
  • Sauty, J. P. (1980). An analysis of hy drodispersive transfer in aquifers. Water Resour. Res. 16 145-158.
  • Sposito, G., Jury, W. A. and Gupta, V. K. (1986). Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and fields. Water Resour. Res. 22 77-99.
  • Sudicky, E. A. (1986). A natural gradient experiment on solute transport in a sand aquifer. Water Resour. Res. 22 2069-2082.
  • Tikhomirov, A. N. (1980). On the rate of convergence in the central limit theorem for weakly dependent random variables. Theory Probab. Appl. 25 800-818.
  • Winter, C. L., Newman, C. M. and Neuman, S. P. (1984). A perturbation expansion for diffusion in a random velocity field. SIAM J. Appl. Math. 44 425-442.