Electronic Journal of Probability

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

Alexander Drewitz

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Parabolic Anderson model Lyapunov exponents intermittency large deviations

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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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