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.
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.
@cite_7
2025859223
A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for v vc. At v=vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditioned by the survival of one individual at a later time T>t. Our numerical results indicate that the size of the population diverges like the exponential of (vc−v)−1 2 in the quasi-stationary regime below vc. Moreover for v>vc, our data indicate that there is no quasi-stationary regime.