Lyapunov exponents for the one-dimensional parabolic Anderson model with drift. (English) Zbl 1191.60079

Summary: 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.


60H25 Random operators and equations (aspects of stochastic analysis)
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
60F10 Large deviations
35R60 PDEs with randomness, stochastic partial differential equations
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