International Journal of Stochastic Analysis

Volume 2011, Article ID 803683, 43 pages

http://dx.doi.org/10.1155/2011/803683

Research Article

## Asymptotics of Negative Exponential Moments for Annealed Brownian Motion in a Renormalized Poisson Potential

^{1}Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA^{2}Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv 01601, Ukraine

Received 24 December 2010; Accepted 6 April 2011

Academic Editor: Nikolai Leonenko

Copyright © 2011 Xia Chen and Alexey Kulik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- A.-S. Sznitman,
*Brownian Motion, Obstacles and Random Media*, Springer Monographs in Mathematics, Springer, Berlin, Germany, 1998. - T. Komorowski, “Brownian motion in a Poisson obstacle field,”
*Séminaire Bourbaki*, vol. 1998/9, no. 853, pp. 91–111, 2000. View at Google Scholar - S. Bezerra, S. Tindel, and F. Viens, “Superdiffusivity for a Brownian polymer in a continuous Gaussian environment,”
*The Annals of Probability*, vol. 36, no. 5, pp. 1642–1675, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. van den Berg, E. Bolthausen, and F. den Hollander, “Brownian survival among Poissonian traps with random shapes at critical intensity,”
*Probability Theory and Related Fields*, vol. 132, no. 2, pp. 163–202, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Germinet, P. D. Hislop, and A. Klein, “Localization for Schrödinger operators with Poisson random potential,”
*Journal of the European Mathematical Society*, vol. 9, no. 3, pp. 577–607, 2007. View at Google Scholar - T. Povel, “Confinement of Brownian motion among Poissonian obstacles in ${\mathbb{R}}^{d}$, $d\ge 3$,”
*Probability Theory and Related Fields*, vol. 114, no. 2, pp. 177–205, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Chen and A. M. Kulik, “Brownian motion and parabolic Anderson model in a renormalized Poisson potential,”
*Annales de l’Institut Henry Poincare*. Accepted. - X. Chen, “Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models,” to appear in
*Annals of Probability*. - P. W. Anderson, “Absence of diffusion in certain random lattices,”
*Physical Review*, vol. 109, no. 5, pp. 1492–1505, 1958. View at Publisher · View at Google Scholar - R. A. Carmona and S. A. Molchanov,
*Parabolic Anderson Problem and Intermittency*, vol. 108 of*Memoirs of the American Mathematical Society*, 1994. View at Zentralblatt MATH - M. Biskup and W. König, “Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model,”
*Journal of Statistical Physics*, vol. 102, no. 5-6, pp. 1253–1270, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. A. Carmona and F. G. Viens, “Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter,”
*Stochastics and Stochastics Reports*, vol. 62, no. 3-4, pp. 251–273, 1998. View at Google Scholar · View at Zentralblatt MATH - M. Cranston, D. Gauthier, and T. S. Mountford, “On large deviations for the parabolic Anderson model,”
*Probability Theory and Related Fields*, vol. 147, no. 1-2, pp. 349–378, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. C. Dalang and C. Mueller, “Intermittency properties in a hyperbolic Anderson problem,”
*Annales de l'Institut Henri Poincaré Probabilités et Statistiques*, vol. 45, no. 4, pp. 1150–1164, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Florescu and F. Viens, “Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space,”
*Probability Theory and Related Fields*, vol. 135, no. 4, pp. 603–644, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Gärtner, F. den Hollander, and G. Maillard, “Intermittency on catalysts: symmetric exclusion,”
*Electronic Journal of Probability*, vol. 12, no. 18, pp. 516–573, 2007. View at Google Scholar · View at Zentralblatt MATH - J. Gärtner and W. König, “Moment asymptotics for the continuous parabolic Anderson model,”
*The Annals of Applied Probability*, vol. 10, no. 1, pp. 192–217, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Gärtner, W. König, and S. A. Molchanov, “Almost sure asymptotics for the continuous parabolic Anderson model,”
*Probability Theory and Related Fields*, vol. 118, no. 4, pp. 547–573, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Gärtner and S. A. Molchanov, “Parabolic problems for the Anderson model. I. Intermittency and related topics,”
*Communications in Mathematical Physics*, vol. 132, no. 3, pp. 613–655, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Stolz, “Non-monotonic random Schrödinger operators: the Anderson model,”
*Journal of Mathematical Analysis and Applications*, vol. 248, no. 1, pp. 173–183, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. A. Pastur, “Spectra of random selfadjoint operators,”
*Uspekhi Matematicheskikh Nauk*, vol. 28, no. 1(169), pp. 3–64, 1973. View at Google Scholar - W. Kirsch and F. Martinelli, “On the density of states of Schrödinger operators with a random potential,”
*Journal of Physics A*, vol. 15, no. 7, pp. 2139–2156, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Lifshitz, “Structure of the energy spectrum of impurity bands in disordered solid solutions,”
*Soviet Physics*, vol. 17, pp. 1159–1170, 1963. View at Google Scholar - I. M. Lifshitz, “Energy spectrum structure and quantum states of disordered condensed systems,”
*Soviet Physics. Uspekhi*, vol. 7, pp. 549–573, 1965. View at Publisher · View at Google Scholar - L. Erdős, “Lifschitz tail in a magnetic field: the nonclassical regime,”
*Probability Theory and Related Fields*, vol. 112, no. 3, pp. 321–371, 1998. View at Publisher · View at Google Scholar - L. Erdős, “Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case,”
*Probability Theory and Related Fields*, vol. 121, no. 2, pp. 219–236, 2001. View at Publisher · View at Google Scholar - R. Fukushima, “Second order asymptotics for Brownian motion among a heavy tailed Poissonian potential,” http://arxiv.org/abs/arXiv:1010.3875.
- D. Hundertmark, W. Kirsch, and S. Warzel, “Classical magnetic Lifshits tails in three space dimensions: impurity potentials with slow anisotropic decay,”
*Markov Processes and Related Fields*, vol. 9, no. 4, pp. 651–660, 2003. View at Google Scholar - O. Khorunzhiy, W. Kirsch, and P. Müller, “Lifshitz tails for spectra of Erdős-Rényi random graphs,”
*The Annals of Applied Probability*, vol. 16, no. 1, pp. 295–309, 2006. View at Publisher · View at Google Scholar - W. Kirsch and F. Martinelli, “On the spectrum of Schrödinger operators with a random potential,”
*Communications in Mathematical Physics*, vol. 85, no. 3, pp. 329–350, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Kirsch and F. Martinelli, “Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians,”
*Communications in Mathematical Physics*, vol. 89, no. 1, pp. 27–40, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Kirsch and B. Metzger, “The integrated density of states for random Schrödinger operators,” in
*Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday*, vol. 76 of*Proc. Sympos. Pure Math.*, pp. 649–696, Amer. Math. Soc., Providence, RI, USA, 2007. View at Google Scholar - W. Kirsch and B. Simon, “Lifshitz tails for periodic plus random potentials,”
*Journal of Statistical Physics*, vol. 42, no. 5-6, pp. 799–808, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Kirsch and S. Warzel, “Lifshits tails caused by anisotropic decay: the emergence of a quantum-classical regime,”
*Mathematical Physics, Analysis and Geometry*, vol. 8, no. 3, pp. 257–285, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Kirsch and S. Warzel, “Anderson localization and Lifshits tails for random surface potentials,”
*Journal of Functional Analysis*, vol. 230, no. 1, pp. 222–250, 2006. View at Google Scholar · View at Zentralblatt MATH - F. Klopp, “Weak disorder localization and Lifshitz tails: continuous Hamiltonians,”
*Annales Henri Poincaré*, vol. 3, no. 4, pp. 711–737, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Klopp and L. Pastur, “Lifshitz tails for random Schrödinger operators with negative singular Poisson potential,”
*Communications in Mathematical Physics*, vol. 206, no. 1, pp. 57–103, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Leschke and S. Warzel, “Quantum-classical transitions in Lifshitz tails with magnetic fields,”
*Physical Review Letters*, vol. 92, no. 8, Article ID 086402, 4 pages, 2004. View at Publisher · View at Google Scholar - S. Nakamura, “Lifshitz tail for Schrödinger operator with random magnetic field,”
*Communications in Mathematical Physics*, vol. 214, no. 3, pp. 565–572, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Ôkura, “An asymptotic property of a certain Brownian motion expectation for large time,”
*Proceedings of the Japan Academy, Series A*, vol. 57, no. 3, pp. 155–159, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. A. Pastur, “The behavior of certain Wiener integrals as $t\to 1$ and the density of states of Schrödinger equations with random potential,”
*Teoreticheskaya i Matematicheskaya Fizika*, vol. 32, no. 1, pp. 88–95, 1977. View at Google Scholar - L. Pastur and A. Figotin,
*Spectra of Random and Almost-Periodic Operators*, vol. 297 of*Grundlehren der Mathematischen Wissenschaften*, Springer, Berlin, Germany, 1992. - B. Simon, “Lifschitz tails for the Anderson model,”
*Journal of Statistical Physics*, vol. 38, no. 1-2, pp. 65–76, 1985. View at Publisher · View at Google Scholar - P. Stollmann, “Lifshitz asymptotics via linear coupling of disorder,”
*Mathematical Physics, Analysis and Geometry*, vol. 2, no. 3, pp. 279–289, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Chen and J. Rosen, “Large deviations and renormalization for Riesz potentials of stable intersection measures,”
*Stochastic Processes and their Applications*, vol. 120, no. 9, pp. 1837–1878, 2010. View at Publisher · View at Google Scholar - M. D. Donsker and S. R. S. Varadhan, “Asymptotics for the Wiener sausage,”
*Communications on Pure and Applied Mathematics*, vol. 28, no. 4, pp. 525–565, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Fikhtengoltz,
*A Course in Differential and Integral Calculus*, Gostekhizdat, Moscow, Russia, 1948. - R. T. Rockafellar,
*Convex Analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, USA, 1970. - H. H. Schaefer,
*Topological Vector Spaces*, Graduate Texts in Mathematics, Springer, New York, NY, USA, 1971. - M. Kac, “On some connections between probability theory and differential and integral equations,” in
*Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability*, pp. 189–215, University of California Press, Berkeley and Los Angeles, 1950. View at Zentralblatt MATH - X. Chen,
*Random Walk Intersections: Large Deviations and Related Topics*, vol. 157 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 2010. - M. D. Donsker and S. R. S. Varadhan, “Asymptotic evaluation of certain Markov process expectations for large time. I. II,”
*Communications on Pure and Applied Mathematics*, vol. 28, pp. 1–47, 1975. View at Google Scholar - J. Feng and T. G. Kurtz,
*Large Deviations for Stochastic Processes*, vol. 131 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 2006. - R. Bass, X. Chen, and J. Rosen, “Large deviations for Riesz potentials of additive processes,”
*Annales de l'Institut Henri Poincaré Probabilités et Statistiques*, vol. 45, no. 3, pp. 626–666, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. A. Adams,
*Sobolev Spaces*, Pure and Applied Mathematics, Vol. 6, Academic Press, London, UK, 1975. - I. C. Gohberg and M. G. Kreĭn,
*Introduction to the Theory of Linear Nonselfadjoint Operators*, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, RI, USA, 1969. - J. L. Doob,
*Stochastic Processes*, John Wiley & Sons, New York, NY, USA, 1953. - R. J. Adler,
*The Geometry of Random Fields*, Wiley Series in Probability and Mathematical Statistic, John Wiley & Sons, Chichester, UK, 1981. - Z. Shi, “Small ball probabilities for a Wiener process under weighted sup-norms, with an application to the supremum of Bessel local times,”
*Journal of Theoretical Probability*, vol. 9, no. 4, pp. 915–929, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. T. Copson,
*Asymptotic Expansions*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 55, Cambridge University Press, New York, NY, USA, 1965.