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On diffusions in media with pockets of large diffusivity

  • Mark Freidlin
  • Leonid KoralovEmail author
  • Alexander Wentzell
Article
  • 21 Downloads

Abstract

We consider diffusion processes in media with pockets of large diffusivity. The asymptotic behavior of such processes is described when the diffusion coefficients in the pockets tend to infinity. The limiting process is identified as a diffusion on the space where each of the pockets is treated as a single point, and certain conditions on the behavior of the process on the boundary of the pockets are imposed. Calculation of various probabilities and expectations related to the limiting process leads to new initial-boundary (and boundary) problems for the corresponding parabolic (and elliptic) PDEs.

Keywords

Non-standard boundary problem Asymptotic problems for diffusion processes and PDEs Long-time influence of small perturbations Convergence of processes 

Mathematics Subject Classification

60F10 35J25 47D07 60J60 

Notes

Acknowledgements

We are very grateful to anonymous referees for making a number of very useful suggestions. While working on this article, M. Freidlin was supported by NSF Grant DMS-1411866 and L. Koralov was supported by NSF Grant DMS-1309084 and ARO Grant W911NF1710419.

References

  1. 1.
    Doob, J.L.: Stochastic Processes. Wiley Classics Library Edition. Wiley, Hoboken (1990)zbMATHGoogle Scholar
  2. 2.
    Dynkin, E.B.: Markov Processes. Springer, Berlin (1965)CrossRefzbMATHGoogle Scholar
  3. 3.
    Freidlin, M.I.: Functional Integration and Partial Differential Equations. Princeton University Press, Princeton (1985)zbMATHGoogle Scholar
  4. 4.
    Freidlin, M.I., Hu, W.: On second order elliptic equations with a small parameter. Commun. Partial Differ. Equ. 38(10), 1712–1736 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Freidlin, M.I., Koralov, L.: Front propagation for reaction-diffusion equations in periodic structures. J. Stat. Phys. (to appear)Google Scholar
  6. 6.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Freidlin, M.I., Koralov, L., Wentzell, A.D.: On the behavior of diffusion processes with traps. Ann. Probab. 45(5), 3202–3222 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Mandl, P.: Analytical Treatment of One-Dimensional Markov Processes. Springer, Berlin (1968)zbMATHGoogle Scholar
  10. 10.
    Maruyama, G., Tanaka, H.: Ergodic property of N-dimensional recurrent Markov processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 13, 157–172 (1959)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Osada, H.: Homogenization of reflecting barrier Brownian motions. In: Pitman Research Notes in Mathematics Series: Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (Sanda/Kyoto, 1990), vol. 283, pp. 59–74. : Longman Scientific & Technical, Longman, England (1993)Google Scholar
  12. 12.
    Tanemura, H.: Homogenization of a reflecting barrier Brownian motion in a continuum percolation cluster in \(\mathbb{R}^d\). Kodai Math. J. 17(2), 228–245 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wentzell, A.D.: On lateral conditions for multidimensional diffusion processes. Teor. Veroyatn. i Primen. 4(2), 172–185 (1959)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsTulane UniversityNew OrleansUSA

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