Skip to main content

Lectures on random media

  • Chapter
  • First Online:
Lectures on Probability Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1581))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Ahn H., Carmona R., Molchanov S. “Parabolic equations with a Levy potential”, Lecture Notes in Control and Information Sciences, 176, Proceed of IFIPWG 7/1 International Conferences, UNCS, 1991, pp. 1–11.

    Google Scholar 

  2. Aizenman M, Molchanov S. “Localization at large disorder and extreme energies: an elementary derivation”, 1993, Comm. Math. Phys, 157, pp245–278.

    Article  MathSciNet  MATH  Google Scholar 

  3. Akhieser N, Glazman A “The theory of linear operators in Hilbert space”, Vol. I, II, Ungar, New York, 1961.

    Google Scholar 

  4. Albeverio S., Surgailis D., Molchanov S. “Stratified structure of the Universe and the Burger equation: a probabilistic approach”, 1993, to appear in Probability theory and related fields.

    Google Scholar 

  5. Alexander K., Molchanov S. “Percolation of the level sets of the random field with a lattice symmetry”, 1994, to appear in Jorn of Stat. Phys.

    Google Scholar 

  6. Anderson P. “Absence of diffusion in certain random lattices”, 1958, Phys. Rev, 109, pp. 1492–1501.

    Article  Google Scholar 

  7. Arnold L., Papanicolaon G., Wihstutz V. “Asymptotic analysis of the Ljapunov's exponents and rotation numbers of the random oscillator and applications” 1986, SIAM J. Appl. Math., Vol. 46.

    Google Scholar 

  8. Avellaneda M., Majda A. “Mathematical models with exact renormalization for turbulent transport (1990), Comm. Math. Phys. V. 131, pp. 381–429.

    Article  MathSciNet  MATH  Google Scholar 

  9. Avellaneda M., Majda A. “Renormalization theory for Eddy Diffusivity in Turbulent Transport”, Phys. Rev. Letters (1992), V. 68, # 20, pp. 3028–3031.

    Article  Google Scholar 

  10. Azbel M., Kaganov M., Lifshitz I. “Electron theory of metals, Consultants Bureau, New York, 1973.

    Google Scholar 

  11. Billingsley “Convergence of Probability measures”, (1968), Willey, New York.

    MATH  Google Scholar 

  12. Bogachev L., Molchanov S. “Mean-field models in the theory of random media I, II, III; Theor. Math. Phys., I (1989), V. 81, II (1990), V. 82, III (1991), V. 87.

    Google Scholar 

  13. Bulycheva O., Molchanov S. “The necessary conditions for the averaging of one-dimensional random media”, Vestnik MGU (Moscow) 1986, #3, pp. 33–38.

    MathSciNet  MATH  Google Scholar 

  14. Carmona R., Lacroix J. “Spectral theory of Random Schrödinger operator”, Birhäuser Verlag, Basel, Boston, Berlin, 1990.

    Book  MATH  Google Scholar 

  15. Caromona R., Molchanov S. “Parabolic Anderson model and intermittency”, Preprint UCI (1992), to appear in “Memoirs of AMS” (1994).

    Google Scholar 

  16. Carmona R., Molchanov S., Noble J. “Parabolic evolution equation with random gaussian potential”, Preprint UCI (1992).

    Google Scholar 

  17. Cycon H., Froese K., Kirsch W., Simon B. “Schrödinger operators with applications to Quantum mechanics and global geometry”, (1987), Springer Verlag, Berlin.

    MATH  Google Scholar 

  18. Cramer H., Lidbetter H. “Stationary random processes”, (1975), Springer-Verlag, Berlin.

    Google Scholar 

  19. Delyon F., Levy Y., Soullard B. “Anderson localization for multidimensional systems at large disorder or low energy”, Comm. Math. Phys. (1985), V. 100, pp. 463–470.

    Article  MathSciNet  MATH  Google Scholar 

  20. Donsker M.D., Varadhan S.R.S. “Asyntotics for the Wiener sausage”, Comm. Pure Appl. Math. (1975), V. 28, pp. 525–565.

    Article  MathSciNet  MATH  Google Scholar 

  21. Von Dreifus H., Klein A. “A new proof of localization in the Anderson tight binding model”, Comm. Math. Phys., (1989), V. 124, pp. 285–299.

    Article  MathSciNet  MATH  Google Scholar 

  22. Durett R. “Probability: theory and examples (1991), Wadsworth and Brooks/Cole Statistics and Probability series”.

    Google Scholar 

  23. Dynkin E.B. “Non-negative eigenfunctions of the Laplace Beltrami operator and browinan motion in certain symmetric space”, (1961), Soviet Math. (Doklady), V. 2, W 6, pp. 1433–1435.

    MATH  Google Scholar 

  24. Ferstenberg H. “Noncommuting random products”, (1963), Trans. Amer. Math. Soc., V. 108, pp. 377–428.

    Article  MathSciNet  Google Scholar 

  25. Freidlin, M. I. “Dirichlet problem for an equations with periodic coefficients”, (1964), Probability theory and Appl. V. 9, pp. 133–139.

    Article  MathSciNet  MATH  Google Scholar 

  26. Freidlin M.I., Wentzell A. D. “Random perturbations of Dynamical Systems”, (1984), Springer-Verlag, Berlin.

    Book  MATH  Google Scholar 

  27. Fröhlich J., Martinelli F., Scoppola E., Spencer T. “Constructive proof of localization in the Anderson tight binding model” (1985), Comm. Math. Phys., V. 101, pp. 21–46. *** DIRECT SUPPORT *** A00I6B38 00004

    Article  MathSciNet  MATH  Google Scholar 

  28. Goldsheid Ya., Molchanov S., Pastur L. “Pure point spectrum of stochastic one dimensional Schrödinger operator”, Func. Anal. and Appl. V. 11, #1 (1977).

    Google Scholar 

    Google Scholar 

  29. Goldsheid Ya., Margulis G. “A condition for simplicity of the spectrum of Ljapunov exponents”, (1987), Sov. Math. Dokl, V. 35, #2, pp. 309–313.

    MathSciNet  MATH  Google Scholar 

  30. Gordon A., Jacsiĉ V., Molchanov S., Simon B., “Spectral properties of random Schrödinger operator with unbounded potential”, 1990, Caltech, preprint, to appear 1993 in Comm. Math. Phys.

    Google Scholar 

  31. Gordon A. “On exceptional value of the boundary phase for the Schrödinger equation on a half-line”, (1992), Russian Mathemat. Surveys, 47, pp. 260–261.

    Article  Google Scholar 

  32. Görtner J., Molchanov S. “Parabolic problems for the Anderson model”, (1990) Comm. Math. Phys., V. 132, pp. 613–655.

    Article  MathSciNet  Google Scholar 

  33. Greven A., den Hollander F. “Branching random walks in random environment: phase transitions for local and global rates, (1992), Probab. Theory Relation Fields.

    Google Scholar 

  34. Greven A., den Hollander F., “Population growth in random media”: I variational formula and phase diagram, II wave front propagation (1991), preprint # 636, University of Heidelberg.

    Google Scholar 

  35. Ibragimov I, Linnik Ju. “Independent and stationary sequences of random variables”. 1971. Wolters-Noordhoff publishing Groningern, Holland.

    MATH  Google Scholar 

  36. Isichenko M. “Percolation, statistical topography and transport in random media”, Rev. Modern Physics, (1992), V. 64, pp. 961–1043.

    Article  MathSciNet  Google Scholar 

  37. Jitomirskaya S., Makarov N., del Rio R., Simon B. “Singular continuous spectrum is generic”, (1993), Caltech preprint, to appear in Bull. AMS.

    Google Scholar 

  38. Kesten, H., “Percolation Theory for Mathematicians” (1982), Bürkhäuser, Boston.

    Book  MATH  Google Scholar 

  39. Kirsh W., Kotani S., Simon B. “Absence of absolutely continuous spectrum for one-dimensional random, but deterministic Schrödinger operators”, Ann. Inst. H. Poincare, (1985), V. 42, p. 383.

    MATH  Google Scholar 

  40. Kirsh W., Molchanov S., Pastur L. “One dimensional Schrödinger operator with unbounded potential: pure point spectrum, I, Funct. Anal. and Appl. (1990), #3, p. 24.

    MathSciNet  Google Scholar 

  41. Kozlov S., “Averaging of random operators”, Mathem. Sbornik, (1979), V. 151, pp. 188–202.

    MathSciNet  Google Scholar 

  42. Kozlov S. “The method of averaging and walks in inhomogeneous environments”, Russian Math. Surveys, (1985), 40: 2, pp. 73–145.

    Article  MATH  Google Scholar 

  43. Kozlov S., Molchanov S. “On conditions under which central limit theorem is applicable to random walk on lattice”, Dokl. Acad. Nauk SSSP (1984), V. 278, pp. 531–534, Sov. Math. Dokl 30 (1984), 410–413.

    MathSciNet  Google Scholar 

  44. Kotani S. “Ljapunov indices determine absolutely continuous spectra of stationary one-dimensional Schrödinger operator”, in Proc. Taneguchi Intern. Symp. on Stochastic Analysis, Katata and Kyoto, (1982), ed. K. Ito, Nort Holland, pp. 225–247.

    Google Scholar 

  45. Kotani S. “Ljapunov exponent and spectra for one-dimensional random Schrödinger operators”, (1986) Proc. Conf on Random Matrices and their Applications, Contemporary Math., V. 50, Providence R.I., pp. 277–286.

    Google Scholar 

  46. Kotani S. “Support theorems for random Schrödinger operators”, Comm. Math. Phys., (1985), V. 97, pp. 443–452.

    Article  MathSciNet  MATH  Google Scholar 

  47. Kotani S., “Absolute continuous spectra for one-dimensional ergodic operators”, (1993), to appear in Proc. Summer Inst. AMS, Cornell, Ithaka, NY.

    Google Scholar 

  48. Kunz H., Souillard B., “The localization transition on the Bethe lattice” (1983), Journ. Phys. Letters, (Paris), V. 44, pp. 411–414.

    Article  Google Scholar 

  49. Lifshitz I., Gredescul S., Pastur L., “Introduction to the theory of disordered media”, (1982), Moscow, Nauka, (1986) Springer-Verlag, Berlin.

    Google Scholar 

  50. McKean H.P., “Stochastic integrals” (1969), Academic press, New York.

    MATH  Google Scholar 

  51. Manakov S., Novikov, S., Pitaevskii, Zakharov V. “The Theory of Solutions, the method of the inverse problem”, Nauka, Moscow, 1980.

    MATH  Google Scholar 

  52. Molchanov S., “The structure of eigenfunctions of one-dimensional disordered systems”, Izv. Acad. Sci. USSR (1978), V. 2, #1, pp. 70–101.

    Google Scholar 

  53. Molchanov S. “Lectures on the localization theory”, (1990), Preprint, Caltech.

    Google Scholar 

  54. Molchanov S. “Intermittency and localization: new results” (1990), Proc. of the Intern. Congr. Math (Kyoto, Japan), Vol. II, pp. 1091–1103.

    MATH  Google Scholar 

  55. Molchanov S. “Ideas in the theory or Random Media”, (1991), Acta. Appl. Math., V. 22, pp. 139–282, Kluver Acad. Publish.

    Article  MathSciNet  MATH  Google Scholar 

  56. Molchanov S., Piterbarg L. “The turbulent diffusion of the temperature gradients”, Dokl. Sov. Acad. Sci. (1986), V. 284, #4.

    Google Scholar 

    Google Scholar 

  57. Molchanov S., Piterbarg L. “Localization of the Rossby topographical waves”, Dokl. Sov. Acad. Sci. (1990), V. 310, #4.

    Google Scholar 

    Google Scholar 

  58. Molchanov S., Ruzmaikin A “Ljapunov exponents and distribution of the magnetic field in dynamo-model” (1993), to appear in “Proc. conference in Probability, theory and Markov processes”, Cornell.

    Google Scholar 

  59. Molchanov S., Piterbarg L. “Heat propagation in random flows” (1992), Russian J. Math. Phys., V. 1, #1, pp. 18–42.

    MathSciNet  Google Scholar 

  60. Mott N., Twose W. “The theory of impurity conduction”, (1961), Adv. Phys., V. 10, pp. 107–163.

    Article  Google Scholar 

  61. Mueller C., Tribe R. “A stochastic PDE arising as the limit of a long range contact processes and its phase transition” (1993), Technical Report, Math. Sci. Inst., Cornell.

    MATH  Google Scholar 

  62. Newman S. “The distribution of Ljapunov exponents: (1986), Comm. Math. Phys., V. 103, pp. 121–126.

    Article  MathSciNet  Google Scholar 

  63. Oseledec V. “A multiplicative ergodic theorem, Ljapunov characteristic numbers in dynamical systems”, Trans. Moscow. Math. Soc. (1968), V. 19, pp. 197–231.

    MathSciNet  Google Scholar 

  64. Piterbarg L. “Dynamics and prediction of the large scale SST anomalies”, (1989) Gidrometeoizdat, Leningrad, Translated in Kluwer (Holland).

    Google Scholar 

  65. Rozovskii B. “Stochastic Differential equations (1991), Kluwer, Holland.

    MATH  Google Scholar 

  66. Ruzmaikin A., Liewer P., Fienman J. “Random cell dynamo” (1993), to appear in Gophys. Astroph. Fluid. Dyn.

    Google Scholar 

  67. Reed M., Simon B. “Methods of Modern Mathematical Physics”, I–IV (1975–1978), Academic Press, New York.

    MATH  Google Scholar 

  68. Simon B., Spencer T. “Trace class perturbation and the absence of absolutely continuous spectrum”, (1989) Comm. Math. Phys., v. 125, pp. 113–125.

    Article  MathSciNet  MATH  Google Scholar 

  69. Simon B., Wolff T. “Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians” (1986), Comm. Pure Appl. Math., V. 39, pp. 75–90.

    Article  MathSciNet  MATH  Google Scholar 

  70. Sinai Ja. “Limit behavior of one-dimensional random walks in random environment”, (1982), Theor. Probab. Appl., 27, pp. 247–258.

    MathSciNet  Google Scholar 

  71. Spitzer F., “Principles of random walk” (1976), Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  72. Sznitman A.S., “Brownian asymptotics in a Poisson environment”, (1991), Preprint ETH-Zentrum (Zürich), to appear in Probab. Theory Relat. Fields.

    Google Scholar 

  73. Papanicolaou G., Varadhan S.R.S. “Boundary value problem with rapidly oscillating random coefficients”, (1981), Coll. Math Soc. Janos Bolyai, 27, Random fields, V. 2 North-Holland, Amersterdam-New York, pp. 835–873.

    MATH  Google Scholar 

  74. Wegner F. “Bounds of the density of states in disordered systems”, (1981) Zeit. Phys. B, Condensed Matter, V. 44, pp. 9–15.

    Article  MathSciNet  Google Scholar 

  75. Zeldovich Ya., Molchanov S., Ruzmaikin A., Sokoloff D. “Intermittency, diffusion and generation in a Non-Stationary Random Medium” (1988), Sov. Sci. Rev., Sec C, Vol. 7, pp. 1–110.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Pierre Bernard

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag

About this chapter

Cite this chapter

Molchanov, S. (1994). Lectures on random media. In: Bernard, P. (eds) Lectures on Probability Theory. Lecture Notes in Mathematics, vol 1581. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073874

Download citation

  • DOI: https://doi.org/10.1007/BFb0073874

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58208-3

  • Online ISBN: 978-3-540-48568-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics