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 Electronic Journal of Probability > Vol. 13 (2008) > Paper 76 open journal systems 


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

Alexander Drewitz, TU Berlin


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.


Full text: PDF

Pages: 2283-2336

Published on: December 21, 2008
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Electronic Journal of Probability. ISSN: 1083-6489