Electronic Journal of Probability

Lyapunov exponents for the one-dimensional parabolic Anderson model with drift

Alexander Drewitz

Full-text: Open access

Abstract

We consider the solution to the one-dimensional parabolic Anderson model with homogeneous initial condition, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for all positive real $p$. These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p$ tends to 0. Furthermore, we show that the solution is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of the solution under the corresponding Gibbs measure. In our context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears as the drift parameter or diffusion constant increase, respectively.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 76, 2283-2336.

Dates
Accepted: 21 December 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819149

Digital Object Identifier
doi:10.1214/EJP.v13-586

Mathematical Reviews number (MathSciNet)
MR2469612

Zentralblatt MATH identifier
1191.60079

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60F10: Large deviations 35B40: Asymptotic behavior of solutions

Keywords
Parabolic Anderson model Lyapunov exponents intermittency large deviations

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Drewitz, Alexander. Lyapunov exponents for the one-dimensional parabolic Anderson model with drift. Electron. J. Probab. 13 (2008), paper no. 76, 2283--2336. doi:10.1214/EJP.v13-586. https://projecteuclid.org/euclid.ejp/1464819149


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