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 space-time potentials W as solutions u to the multiplicative stochastic heat equation u(t, x) = 1 + ∫ t 0 κ∆u(s, x)ds+ ∫ t 0 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 = limt→∞ 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 non-trivial 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/(2H+1) .