## The Annals of Probability

### LongTime Tails in The Parabolic Anderson Model with Bounded Potential

#### Abstract

We consider the parabolic Anderson problem $\partial_t u = \kappa\Delta u + \xi u$ on $(0,\infty) \times \mathbb{Z}^d$ with random i.i.d. potential $\xi = (\xi(z))_{z\in \mathbb{Z}^d}$ and the initial condition $u(0,\cdot) \equiv 1$. Our main assumption is that $\esssup \xi(0)=0$. Depending on the thickness of the distribution $\Prob (\xi(0) \in \cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t, 0)$ and the almost­sure asymptotics of $u(t, 0)$ as $t \to \infty$ in terms of variational problems. As a by­product, we establish Lifshitz tails for the random Schrödinger operator $-\kappa \Delta - \xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.

#### Article information

Source
Ann. Probab., Volume 29, Number 2 (2001), 636-682.

Dates
First available in Project Euclid: 21 December 2001

https://projecteuclid.org/euclid.aop/1008956688

Digital Object Identifier
doi:10.1214/aop/1008956688

Mathematical Reviews number (MathSciNet)
MR1849173

Zentralblatt MATH identifier
1018.60093

#### Citation

Biskup, Marek; and König, Wolfgang. LongTime Tails in The Parabolic Anderson Model with Bounded Potential. Ann. Probab. 29 (2001), no. 2, 636--682. doi:10.1214/aop/1008956688. https://projecteuclid.org/euclid.aop/1008956688

#### References

• Antal, P. (1994). Trapping problems for the simple random walk. Ph.D. dissertation, ETH.
• Antal, P. (1995). Enlargement of obstacles for the simple random walk. Ann. Probab. 23 1061- 1101.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press.
• Biskup M. and K ¨onig W. (1998). On a variational problem related to the one-dimensional parabolic Anderson model. Unpublished manuscript.
• Biskup, M. and K ¨onig, W. (2000). Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model. J. Statist. Phys. To appear.
• Buffet, E. and Pul´e, J. V. (1997). A model of continuous polymers with random charges. J. Math. Phys. 38 5143-5152.
• Carmona, R. and Lacroix, J. (1990). Spectral Theory of Random Schr¨odinger Operators. Birkh¨auser, Boston.
• Carmona, R. and Molchanov, S. A. (1994). Parabolic Anderson Problem and Intermittency. Amer. Math. Soc., Providence, RI.
• Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston.
• Donsker, M. and Varadhan, S. R. S. (1975). Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 525-565.
• Donsker, M. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721-747.
• Gandolfi, A., Keane, M. and Russo, L. (1988). On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16 1147-1157.
• G¨artner, J. and den Hollander, F. (1999). Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 1-54.
• G¨artner, J. and K ¨onig, W. (2000). Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10 192-217.
• G¨artner, J., K ¨onig, W. and Molchanov, S. (1999). Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Related Fields. To appear.
• G¨artner, J., K ¨onig, W. and Molchanov, S. (2001). Parabolic problems for the Anderson model. III. Contribution from high peaks. Unpublished manuscript.
• G¨artner, J. and Molchanov, S. (1990). Parabolic problems for the Anderson model I. Intermittency and related topics. Comm. Math. Phys. 132 613-655.
• G¨artner, J. and Molchanov, S. (1998). Parabolic problems for the Anderson model. II. Secondorder asymptotics and structure of high peaks. Probab. Theory Related Fields 111 17-55.
• G¨artner, J. and Molchanov, S. (2000). Moment asymptotics and Lifshitz tails for the parabolic Anderson model. In Stochastic Models. Proceedings of the International Conference on Stochastic Models. Amer. Math. Soc., Providence, RI.
• Grimmett, G.R. (1989). Percolation. Springer, Berlin.
• Lieb, E.H. and Loss, M. (1997). Analysis. Amer. Math. Soc., Providence, RI.
• Merkl, F. and W ¨uthrich, M. (2000). Infinite volume asymptotics of the ground state energy in a scaled Poissonian potential. Unpublished manuscript.
• Sznitman, A.-S. (1998). Brownian motion, Obstacles and Random Media. Springer, Berlin.