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

Drewitz, Alexander

Electron. J. Probab. 13, 2283-2336 (2008).

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.

MSC:

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

Keywords:

parabolic anderson model; Lyapunov exponents; intermittency; large deviations