Anomalous slow diffusion from perpetual homogenization

Annals of Probability

Volumn 31, Issue 4, 2003, Pages 1935-1969

  Owhadi, Houman ^a  

^a AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

Anomalous diffusion; Davies' conjecture; Diffusion on fractal media; Exponential martingale inequality; Heat kernel; Multi scale homogenization; Periodic operator; Subharmonic

Indexed keywords

EID: 0347593392     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1068646372     Document Type: Article

References (38)

1. ALLAIRE, G. and BRIANE, M. (1996). Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh Sec. A 126 297-342.
2. ANCONA, A. (1997). First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains. J. d'Analyse Math. 72 45-91.
3. AVELLANEDA, M. (1987). Iterated homogenization, differential effective medium theory and applications. Comm. Pure Appl. Math. XL 527-554.
4. AVELLANEDA, M. (1996). Homogenization and renormalization, the mathematics of multi-scale random media and turbulent diffusion. In Dynamical Systems and Probabilistic Methods in Partial Differential Equations (P. Deift, C. D. Levermore and C. E. Wayne, eds.) 251-268. Amer. Math. Soc., Providence, RI.
5. AVELLANEDA, M. and MAJDA, A. (1990). Mathematical models with exact renormalization for turbulent transport. Comm. Math. Phys. 131 381-429.
6. BARLOW, M. T. (1998). Diffusions on Fractals. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1781 1-121. Springer, New York.
7. BARLOW, M. T. and BASS, R. F. (1999). Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 54 673-744.
8. BEN AROUS, G. and OWHADI, H. (2001). Multi-scale homogenization with bounded ratios and anomalous slow diffusion. Unpublished manuscript. Preprint available at www.cmi.univ-mrs.fr/̃owhadi/.
9. BEN AROUS, G. and OWHADI, H. (2002). Super-diffusivity in a shear flow model from perpetual homogenization. Comm. Math. Phys. 227 281-302.
10. BHATTACHARYA, R. (1999). Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. Ann. Appl. Probab. 9 951- 1020.
11. BRUGGERMAN, D. A. G. (1935). Berechnung verschiedener physkalisher konstanten von heterogenen Substanzen. Ann. Phys. 24 634.
12. CHEN, Z. Q., QIAN, Z., Hu, Y. and ZHENG, W. (1998). Stability and approximations od symmetric diffusion semigroups and kernels. J. Funct. Anal. 152 255-280.
13. DAVIES, E. B. (1993). Heat kernels in one dimension. Quart. J. Math. Oxford 44 283-299.
14. DEMBO, A. (1996). Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab. 1 11-17 (electronic).
15. ELLIS, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York.
16. FANNJIANG, A. and KOMOROWSKI, T. (2000). Fractional Brownian motion limit for motions in turbulence. Ann. Appl. Probab. 10 1100-1120.
17. FANNJIANG, A. and PAPANICOLAOU, G. C. (1994). Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 333-408.
18. GARLAND, J. C. and TANNER, D. B. (1977). AIP Conference Proceedings. Amer. Inst. Phys., New York.
19. GILBARG, D. and TRUDINGER, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, New York.
20. GLIMM, J. and ZHANG, Q. (1992). Inertial range scaling of laminar shear flow as a model of turbulent transport. Comm. Math. Phys. 146 217-229.
21. GRÜTER, M. and WIDMAN, K.-O. (1982). The Green function for uniformly elliptic equations. Manuscripta Math. 37 303-342.
22. HAMBLY, B. M., KUMAGAI, T., KUSUOKA, S. and ZHOU, X. Y. (1998). Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets. Preprint.
23. JIKOV, V. V. and KOZLOV, S. M. (1999). Multiscaled homogenization. In Homogenization: Serguei Kozlov Memorial Volume (V. Berdichevsky, V. Jikov and G. Papanicolaou, eds.) 35-64. World Scientific, Singapore.
24. KELLER, G. (1998). Equilibrium States in Ergodic Theory. Cambridge Univ. Press.
25. KOZLOV, S. (1995). Multiscaled approach in homogenization. In Proceeding of the Second Workshop on Composite Media and Homogenization Theory 217-229. World Scientific, Singapore.
26. NORRIS, A. N. (1985). A differential scheme for the effective moduli of composites. Mechanics of Materials 4 1-16.
27. NORRIS, J. R. (1997). Long-time behaviour of heat flow: Global estimates and exact asymptotics. Arch. Rational Mech. Anal. 140 161-195.
28. OLLA, S. (1994). Homogenization of diffusion processes in random fields. Ecole Polytechnique. Cours Ecole Polytechnique.
29. OSADA, H. (1995). Self-similar diffusions on a class of infinitely ramified fractals. J. Math. Soc. Japan 47 591-616.
30. OWHADI, H. (2001 a). Anomalous diffusion and homogenization on an infinite number of scales. Ph.D. dissertation, EPFL, Swiss Federal Institute of Technology. Available at dmawww.epfl.ch/̃owhadi/.
31. OWHADI, H. (2001b). Super-diffusion in turbulence: Kolmogorov Richardson laws versus Landau Lagrangian chaos. To appear.
32. n. Differential and Integral Equations 5 481-493.
33. RICHARDSON, L. F. (1926). Atmosphere diffusion shown on a distance- neighbour graph. Proc. Roy. Soc. London Ser. A 110 709.
34. RUELLE, D. (1978). Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics. Addison-Wesley, Reading, MA.
35. SEIGNOUREL, P. (1998). Processus dans un milieu irregulier. Une approche par les formes de Dirichlet. Preprint, Ecole Polytechnique, France.
36. SEN, P. N., SCALA, C. and COHEN, M. H. (1981). Geophysics 46 781.
37. STAMPACCHIA, G. (1965). Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 189-258.
38. STAMPACCHIA, G. (1966). Equations elliptiques du second ordre à coeffici discontinus. Univ. Montréal Press.