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 + ∫0t 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 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).
All Science Journal Classification (ASJC) codes
- Modelling and Simulation