Advertisement

Lectures on random media

  • S. Molchanov
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1581)

Keywords

Random Walk Anderson Model Stochastic Partial Differential Equation Stokes Drift Localization Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [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. [2]
    Aizenman M, Molchanov S. “Localization at large disorder and extreme energies: an elementary derivation”, 1993, Comm. Math. Phys, 157, pp245–278.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Akhieser N, Glazman A “The theory of linear operators in Hilbert space”, Vol. I, II, Ungar, New York, 1961.Google Scholar
  4. [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. [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. [6]
    Anderson P. “Absence of diffusion in certain random lattices”, 1958, Phys. Rev, 109, pp. 1492–1501.CrossRefGoogle Scholar
  7. [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. [8]
    Avellaneda M., Majda A. “Mathematical models with exact renormalization for turbulent transport (1990), Comm. Math. Phys. V. 131, pp. 381–429.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Avellaneda M., Majda A. “Renormalization theory for Eddy Diffusivity in Turbulent Transport”, Phys. Rev. Letters (1992), V. 68, # 20, pp. 3028–3031.CrossRefGoogle Scholar
  10. [10]
    Azbel M., Kaganov M., Lifshitz I. “Electron theory of metals, Consultants Bureau, New York, 1973.Google Scholar
  11. [11]
    Billingsley “Convergence of Probability measures”, (1968), Willey, New York.zbMATHGoogle Scholar
  12. [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. [13]
    Bulycheva O., Molchanov S. “The necessary conditions for the averaging of one-dimensional random media”, Vestnik MGU (Moscow) 1986, #3, pp. 33–38.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Carmona R., Lacroix J. “Spectral theory of Random Schrödinger operator”, Birhäuser Verlag, Basel, Boston, Berlin, 1990.CrossRefzbMATHGoogle Scholar
  15. [15]
    Caromona R., Molchanov S. “Parabolic Anderson model and intermittency”, Preprint UCI (1992), to appear in “Memoirs of AMS” (1994).Google Scholar
  16. [16]
    Carmona R., Molchanov S., Noble J. “Parabolic evolution equation with random gaussian potential”, Preprint UCI (1992).Google Scholar
  17. [17]
    Cycon H., Froese K., Kirsch W., Simon B. “Schrödinger operators with applications to Quantum mechanics and global geometry”, (1987), Springer Verlag, Berlin.zbMATHGoogle Scholar
  18. [18]
    Cramer H., Lidbetter H. “Stationary random processes”, (1975), Springer-Verlag, Berlin.Google Scholar
  19. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Donsker M.D., Varadhan S.R.S. “Asyntotics for the Wiener sausage”, Comm. Pure Appl. Math. (1975), V. 28, pp. 525–565.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Durett R. “Probability: theory and examples (1991), Wadsworth and Brooks/Cole Statistics and Probability series”.Google Scholar
  23. [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.zbMATHGoogle Scholar
  24. [24]
    Ferstenberg H. “Noncommuting random products”, (1963), Trans. Amer. Math. Soc., V. 108, pp. 377–428.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Freidlin, M. I. “Dirichlet problem for an equations with periodic coefficients”, (1964), Probability theory and Appl. V. 9, pp. 133–139.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Freidlin M.I., Wentzell A. D. “Random perturbations of Dynamical Systems”, (1984), Springer-Verlag, Berlin.CrossRefzbMATHGoogle Scholar
  27. [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 00004MathSciNetCrossRefzbMATHGoogle Scholar
  28. [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
  29. [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.MathSciNetzbMATHGoogle Scholar
  30. [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. [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.CrossRefGoogle Scholar
  32. [32]
    Görtner J., Molchanov S. “Parabolic problems for the Anderson model”, (1990) Comm. Math. Phys., V. 132, pp. 613–655.MathSciNetCrossRefGoogle Scholar
  33. [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. [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. [35]
    Ibragimov I, Linnik Ju. “Independent and stationary sequences of random variables”. 1971. Wolters-Noordhoff publishing Groningern, Holland.zbMATHGoogle Scholar
  36. [36]
    Isichenko M. “Percolation, statistical topography and transport in random media”, Rev. Modern Physics, (1992), V. 64, pp. 961–1043.MathSciNetCrossRefGoogle Scholar
  37. [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. [38]
    Kesten, H., “Percolation Theory for Mathematicians” (1982), Bürkhäuser, Boston.CrossRefzbMATHGoogle Scholar
  39. [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.zbMATHGoogle Scholar
  40. [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.MathSciNetGoogle Scholar
  41. [41]
    Kozlov S., “Averaging of random operators”, Mathem. Sbornik, (1979), V. 151, pp. 188–202.MathSciNetGoogle Scholar
  42. [42]
    Kozlov S. “The method of averaging and walks in inhomogeneous environments”, Russian Math. Surveys, (1985), 40: 2, pp. 73–145.CrossRefzbMATHGoogle Scholar
  43. [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.MathSciNetGoogle Scholar
  44. [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. [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. [46]
    Kotani S. “Support theorems for random Schrödinger operators”, Comm. Math. Phys., (1985), V. 97, pp. 443–452.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [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. [48]
    Kunz H., Souillard B., “The localization transition on the Bethe lattice” (1983), Journ. Phys. Letters, (Paris), V. 44, pp. 411–414.CrossRefGoogle Scholar
  49. [49]
    Lifshitz I., Gredescul S., Pastur L., “Introduction to the theory of disordered media”, (1982), Moscow, Nauka, (1986) Springer-Verlag, Berlin.Google Scholar
  50. [50]
    McKean H.P., “Stochastic integrals” (1969), Academic press, New York.zbMATHGoogle Scholar
  51. [51]
    Manakov S., Novikov, S., Pitaevskii, Zakharov V. “The Theory of Solutions, the method of the inverse problem”, Nauka, Moscow, 1980.zbMATHGoogle Scholar
  52. [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. [53]
    Molchanov S. “Lectures on the localization theory”, (1990), Preprint, Caltech.Google Scholar
  54. [54]
    Molchanov S. “Intermittency and localization: new results” (1990), Proc. of the Intern. Congr. Math (Kyoto, Japan), Vol. II, pp. 1091–1103.zbMATHGoogle Scholar
  55. [55]
    Molchanov S. “Ideas in the theory or Random Media”, (1991), Acta. Appl. Math., V. 22, pp. 139–282, Kluver Acad. Publish.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    Molchanov S., Piterbarg L. “The turbulent diffusion of the temperature gradients”, Dokl. Sov. Acad. Sci. (1986), V. 284, #4.Google Scholar
  57. [57]
    Molchanov S., Piterbarg L. “Localization of the Rossby topographical waves”, Dokl. Sov. Acad. Sci. (1990), V. 310, #4.Google Scholar
  58. [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. [59]
    Molchanov S., Piterbarg L. “Heat propagation in random flows” (1992), Russian J. Math. Phys., V. 1, #1, pp. 18–42.MathSciNetGoogle Scholar
  60. [60]
    Mott N., Twose W. “The theory of impurity conduction”, (1961), Adv. Phys., V. 10, pp. 107–163.CrossRefGoogle Scholar
  61. [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.zbMATHGoogle Scholar
  62. [62]
    Newman S. “The distribution of Ljapunov exponents: (1986), Comm. Math. Phys., V. 103, pp. 121–126.MathSciNetCrossRefGoogle Scholar
  63. [63]
    Oseledec V. “A multiplicative ergodic theorem, Ljapunov characteristic numbers in dynamical systems”, Trans. Moscow. Math. Soc. (1968), V. 19, pp. 197–231.MathSciNetGoogle Scholar
  64. [64]
    Piterbarg L. “Dynamics and prediction of the large scale SST anomalies”, (1989) Gidrometeoizdat, Leningrad, Translated in Kluwer (Holland).Google Scholar
  65. [65]
    Rozovskii B. “Stochastic Differential equations (1991), Kluwer, Holland.zbMATHGoogle Scholar
  66. [66]
    Ruzmaikin A., Liewer P., Fienman J. “Random cell dynamo” (1993), to appear in Gophys. Astroph. Fluid. Dyn.Google Scholar
  67. [67]
    Reed M., Simon B. “Methods of Modern Mathematical Physics”, I–IV (1975–1978), Academic Press, New York.zbMATHGoogle Scholar
  68. [68]
    Simon B., Spencer T. “Trace class perturbation and the absence of absolutely continuous spectrum”, (1989) Comm. Math. Phys., v. 125, pp. 113–125.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    Sinai Ja. “Limit behavior of one-dimensional random walks in random environment”, (1982), Theor. Probab. Appl., 27, pp. 247–258.MathSciNetGoogle Scholar
  71. [71]
    Spitzer F., “Principles of random walk” (1976), Springer-Verlag, New York.CrossRefzbMATHGoogle Scholar
  72. [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. [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.zbMATHGoogle Scholar
  74. [74]
    Wegner F. “Bounds of the density of states in disordered systems”, (1981) Zeit. Phys. B, Condensed Matter, V. 44, pp. 9–15.MathSciNetCrossRefGoogle Scholar
  75. [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.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • S. Molchanov
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos Angeles

Personalised recommendations