Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients
Download PDF
Download PDF
  • Published: March 1988

Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients

  • Hisao Watanabe1 

Probability Theory and Related Fields volume 77, pages 359–378 (1988)Cite this article

Summary

In this paper we consider the parabolic equation with random coefficients:

$$D_t u^\varepsilon (t,x) = \sum\limits_{ij} {a_{ij} } \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)D_{x_i x_j } u^\varepsilon (t\user2{,}x) + \sum\limits_i {b_i } \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)D_{x_i } u^\varepsilon (t\user2{,}x).$$

We show that uε(t,x) converges to the solution uo(t,x) of the averaging equation:

$$D_t u^0 (t,x) = \sum\limits_{ij} E \left( {a_{ij} \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)} \right)D_{x_i x_j } u^0 (t\user2{,}x) + \sum\limits_i {E\left( {b_i \left( {\frac{t}{\varepsilon }\user2{,}x,\omega } \right)} \right)} D_{x_i } u^0 (t\user2{,}x).$$

Also, the fluctuation process \(y^\varepsilon (t,x)\left( { \equiv \left( {u^\varepsilon (t,x)--u^0 (t\user2{,}x)} \right)/\sqrt \varepsilon } \right)\) converges weakly to a generalized Ornstein-Uhlenbeck process on L′.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic analysis for periodic structure, pp. 516–533. Amsterdam: North-Holland 1978

    Google Scholar 

  2. Brodskiy, Ya.S., Lakacher, B.Ya.: Fluctuation in averaging scheme for differential equations with random right hand side. (Russian) Theory of random process 12, 8–17 (1984)

    Google Scholar 

  3. Davydov, Yu.A.: Convergence of distributions generated by stationary stochastic processes. Theory Probab. Appl. 13, 691–696 (1968)

    Google Scholar 

  4. Eidel'man, S.D.: Parabolic system. Amsterdam: North-Holland 1969

    Google Scholar 

  5. Friedman, A.: Partial differential equations of parabolic type. Prentice Hall, Englewood Cliffs, N.J., 1964

    Google Scholar 

  6. Geman, S.: A method of averaging for random differential equations with applications to stability and stochastic approximations. In: Bharuch Reid, A.T. (ed.) Approximate solution of random equation, pp. 37–85, New York: North-Holland 1979

    Google Scholar 

  7. Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables. Groningen: Wolter-Noordhoff 1971

    Google Scholar 

  8. Itô, K.: Infinite dimensional Ornstein-Uhlenbeck processes in stochastic analysis. pp. 197–224. Tokyo: Kinokuniya 1984

    Google Scholar 

  9. Kesten, H., Papanicolaou, G.C.: A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128 (1979)

    Google Scholar 

  10. Khas'minskii, R.Z.: Principle of averaging for parabolic and elliptic differential equations and for Markov processes with small diffusion. Theory Probab. Appl. 8, 1–21 (1963)

    Google Scholar 

  11. Khas'minskii, R.Z.: On the stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11, 211–222 (1966)

    Google Scholar 

  12. Krylov, N.M., Bogoliubov, N.N.: Introduction to nonlinear mechanics. Princeton: Princeton University Press 1947

    Google Scholar 

  13. Kushner, H.J.: A martingale method for the convergence of a sequence of processes to a jumpdiffusion process. Z. Wahrscheinlichkeitstheor. Verw. geb. 53, 209–219 (1980)

    Google Scholar 

  14. Mitoma, I.: On the sample continuity of L′-processes. J. Math. Soc. Japan 35, 629–636 (1983)

    Google Scholar 

  15. Mitoma, I.: Tightness of probabilities on C([0, 1]; L′) and D([0, 1]; L′). Ann. Probab. 11, 989–999 (1983)

    Google Scholar 

  16. Nash, Jr., J.F.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Google Scholar 

  17. Porper, F.O., Eidel'man, S.D.: Two-sided estimates of fundamental solutions of second-order parabolic equation and some applications. Russ. Math. Surv. 39, 119–178 (1984)

    Google Scholar 

  18. Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  19. Watanabe, H.: Fluctuations in certain dynamical systems with averaging. Stochastic Processes Appl. 21, 147–157 (1985)

    Google Scholar 

  20. Watanabe, H.: Averaging and fluctuations of certain stochastic equations. In: Albeverio, S., Blanchard, Ph., Streit, L. (eds.) Stochastic Processes Mathematics and Physics. Bielefeld 1985. (Lect. Notes Math., vol. 1250, pp. 348–355) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  21. Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Averaging of parabolic operators. Trans. Mosc. Math. Soc. 45, 189–241 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Science, Faculty of Engineering, Kyushu University, 812, Fukuoka, Japan

    Hisao Watanabe

Authors
  1. Hisao Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Watanabe, H. Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients. Probab. Th. Rel. Fields 77, 359–378 (1988). https://doi.org/10.1007/BF00319294

Download citation

  • Received: 12 December 1985

  • Revised: 03 November 1987

  • Issue Date: March 1988

  • DOI: https://doi.org/10.1007/BF00319294

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Parabolic Equation
  • Average Equation
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature