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.
"Moments and Lyapunov exponents for the parabolic Anderson model." Ann. Appl. Probab. 24 (3) 1172 - 1198, June 2014. https://doi.org/10.1214/13-AAP944