Borodin, Alexei et Corwin, Ivan, Moments and Lyapunov exponents for the parabolic Anderson model, Annals of applied probability , 24(3), 2014, pp. 1172-1198
We study the parabolic Anderson model in (1+1) dimensions with nearest neighbor jumps and space.time white noise (discrete space/continuous time). We prove a contour integral formula for the second moment and compute the second moment Lyapunov exponent. For the model with only jumps to the right, we prove a contour integral formula for all moments and compute moment Lyapunov exponents of all orders.