LYAPUNOV EXPONENTS FOR STOCHASTIC ANDERSON MODELS WITH NON-GAUSSIAN NOISE

Abstract

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 [Formula: see text] 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 β2κ-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).