Lyapunov exponents for stochastic Anderson models with non-gaussian noise

The stochastic Anderson model in discrete or continuous space is defined for a class of non-Gaussian spacetime potentials W as solutions u to the multiplicative stochastic heat equation u(t,x)=1+ ∫₀ ^t κΔ u(s,x)ds + ∫₀^t βW (ds, x) u(s, x) with diffusivity κ and inverse-temperature β. The relation with the corresponding polymer model in a random environment is given. The large time exponential behavior of u is studied via its almost sure Lyapunov exponent λ = lim_t→∞ t^-1 log u(t, x), which is proved to exist, and is estimated as a function of β and κ for β² κ^-1 bounded below: positivity and nontrivial upper bounds are established, generalizing and improving existing results. In discrete space λ is of order β²/log (β²/κ) and in continuous space it is between β² (κ/β²) ^H/(H+1) and β² (κ/β²) ^H/(1+3H).