Summary
This paper is devoted to the analysis of the large time behavior of the solutions of the Anderson parabolic problem:
when the potential ξ(x) is a homogeneous ergodic random field on ℝd. Our goal is to prove the asymptotic spatial intermittency of the solution and for this reason, we analyze the large time properties of all the moments of the positive solutions. This provides an extension to the continuous space ℝd of the work done originally by Gärtner and Molchanov in the case of the lattice ℤd. In the process of our moment analysis, we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized Gaussian fields, interpolating continuously between the exponent 2 which is found in the case of bona fide continuous Gaussian fields ξ(x) and the exponent 3/2 appearing in the case of a one dimensional white noise. Finally, we also determine the precise almost sure large time asymptotics of the positive solutions.
References
Adler, R.J.: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS Lect. Notes # 12. Hayward (1990)
Ahn, H.S., Carmona, R., Molchanov, S.A.: Parabolic equations with a Lévy random potential. Proc. Charlotte Conf. on Stochast. Part. Diff. Equations, 1991 (to appear)
Aizenman, M., Molchanov, S.A.: Localization at large disorder and extreme energies: an elementary derivation (preprint, 1993)
Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Boston: Birkhaüser 1990
Carmona, R., Molchanov, S.A.: Intermittency and phase transitions for some particle systems in random media. Proc. Katata Symp, June 1990 (to appear)
Carmona, R., Molchanov, S.A.: Parabolic Anderson problem and intermittency. Memoirs Amer. Math. Soc.108 #514 (1994)
Carmona, R., Grishin, S., Molchanov, S.A.: Asymptotic for the boundary parabolic Anderson problem in a half space (preprint, 1994).
Donsker, M.D., Varadhan, S.R.S.: Asymtotics for the Wiener saussage. Commun. Pure and Appl. Math28, 525–565 (1975)
Donsker, M., Varadhan, S.R.S.: Asymptotics for the Polaron. Commun. Pure Appl. Math.36, 183–212 (1983)
Fernique, X.: Regularité des trajectoires des fonctions aléatoires gaussiennes. (Lect. Notes Math., Vol.480, 1–96 (1975)
Figotin, A., Pastur, L.A.: Spectral properties of disordered systems in the one body approximation. New York: Springer 1993
Gärtner, J., Molchanov, S.A.: Parabolic problems for the Anderson model. Comm. Math. Phys.132, 613–655 (1990)
Guelfand, I.M., Vilenkin, G.: Generalized Functions, Vol. 4. New York: Academic Press 1964
Klein A., Landau L.J.: Construction of a unique self-adjoint generator for a symmetric local semigroup. J. Funct. Anal.44, 121–137 (1981)
Lifschitz, I.M., Gredeskul, S.A., Pastur, L.A.: Introduction to the theory of disordered systems. New York: Wiley 1988
Mansmann, U.: Strong coupling limit for a certain class of polaron models. Probab. Theory Relat. Fields90, 427–446 (1991)
Molchanov, S.A.: Ideas in the theory of random media. Acta Applicandae Math.22, 139–282 (1990)
Reed, M., Simon, B.: Methods of modern mathematical physics II: Fourier analysis, self-adjointness. New York: Academic Press 1975
Sznitman, A.S.: Brownian asymptotics in a Poissonian environment. Probab. Theory Relat. Fields95, 155–174 (1993)
Weber, M.: Sur le comportement Asymptotique des Processus Gaussiens Stationaires. in Aspects Statistiques et Aspects Physiques dse Processus Gaussiens. Paris: CNRS pp. 563–567 1981
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Partially supported by ONR N00014-91-1010
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Carmona, R.A., Molchanov, S.A. Stationary parabolic Anderson model and intermittency. Probab. Th. Rel. Fields 102, 433–453 (1995). https://doi.org/10.1007/BF01198845
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DOI: https://doi.org/10.1007/BF01198845
Mathematics Subject Classification
- 60H25