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.
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