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Anderson Parabolic Model for a Quasi-Stationary Medium

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Abstract

We study the Anderson Parabolic Model for a random medium which is a product of an i.i.d. space-like random field and a white noise. The model has long range space-time correlations and is intermediate between the stationary case and the “turbulent” one, which were studied in previous works. Under some natural assumptions on the distribution of the space potential, we prove existence and uniqueness, and derive the long time asymptotics for the annealed moments, and the “semi-annealed” ones, for which expectation is taken only w.r.t. the white noise. A conjecture for the fully quenched case is discussed on a simplified model.

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References

  1. Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. I. Intermittency and related topics. Commun. Math. Phys. 132(3), 613–655 (1990)

    Article  MATH  ADS  Google Scholar 

  2. Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. II. Second-order asymptotics and structure of high peaks. Probab. Theory Related Fields 111(1), 17–55 (1998)

    Article  MATH  Google Scholar 

  3. Zeldovich, Ya., Molchanov, S., Ruzmaikin, A., Sokolov, D.: Intermittency, Diffusion and Generation in a Non-Stationary Random Medium. Russian Reviews in Mathematical Physics. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  4. Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Mem. Am. Math. Soc. 108, 518 (1994)

    Google Scholar 

  5. Molchanov, S.: Lectures on Random Media. In: Bernard, P. (ed.) Lectures in Probability Theory, Ecole d’ Eté de Probabilités de Saint Flour XXII, 1992. Lecture Notes in Mathematics, vol. 1581, pp. 242–411. Springer, Berlin (1994)

    Google Scholar 

  6. Boldrighini, C., Minlos, R.A., Pellegrinotti, A.: Random walks in quenched i.i.d. space–time random environment are always a.s. diffusive. Probab. Theory Relat. Fields 129(1), 133–156 (2004)

    Article  MATH  Google Scholar 

  7. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and Applications (No. 27). Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  8. Ben Arous, G., Bogachev, L., Molchanov, S.A.: Limit theorems for sums of random exponentials. Probab. Theory Relat. Fields 132, 579–612 (2005)

    Article  MATH  Google Scholar 

  9. Ross, S.: Introduction to Probability Models. Academic Press, Orlando (1985)

    MATH  Google Scholar 

  10. Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185, Rome, Italy

    C. Boldrighini

  2. Department of Mathematics and Statistics, University of North Carolina at Charlotte, 376 Fretwell Bldg. 9201 University City Blvd., Charlotte, NC, 28223-0001, USA

    S. Molchanov

  3. Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1, 00146, Rome, Italy

    A. Pellegrinotti

Authors
  1. C. Boldrighini
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  2. S. Molchanov
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  3. A. Pellegrinotti
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Correspondence to C. Boldrighini.

Additional information

C. Boldrighini and A. Pellegrinotti are partially supported by INdAM (GNFM) and MURST research founds.

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Boldrighini, C., Molchanov, S. & Pellegrinotti, A. Anderson Parabolic Model for a Quasi-Stationary Medium. J Stat Phys 129, 151–169 (2007). https://doi.org/10.1007/s10955-007-9364-3

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  • DOI: https://doi.org/10.1007/s10955-007-9364-3

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