We consider the solution @math to the one-dimensional parabolic Anderson model with homogeneous initial condition @math , 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 @math -th annealed Lyapunov exponents for all @math These results enable us to prove the heuristically plausible fact that the @math -th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as @math Furthermore, we show that @math is @math -intermittent for @math large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of @math under the corresponding Gibbs measure. In this context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears.
@cite_7: Let (η n ) be the infinite particle system on ℤ whose evolution is as follows. At each unit of time each particle independently is replaced by a new generation. The size of a new generation descending from a particle at sitex has distributionF x and each of its members independently jumps to sitex±1 with probability (1±h) 2,h∈[0, 1]. The sequence F x is i.i.d. with uniformly bounded second moment and is kept fixed during the evolution. The initial configurationη0 is shift invariant and ergodic.
A situation similar in spirit to ours has been examined in the seminal article @cite_7 by Greven and den Hollander, motivated from the point of view of population dynamics. The model treated there is a discrete-time branching model in random environment with drift and corresponds to the case of a bounded i.i.d. potential. In particular, their motivation stems from the discrete-time analogue described at the end of section . The results are formulated by the use of nontrivial variational formulae. While the authors concentrate on the explicit dependence of the results on the drift parameter @math an advantage of our approach is that we may compute the @math -th annealed Lyapunov exponents for all @math and characterise them in a simpler way.