Small Kappa Asymptotics of the Almost Sure Lyapunov Exponent for the Continuum Parabolic Anderson Model

We prove that the almost sure Lyapunov exponent \lambda(\kappa) of the continuous space Parabolic Anderson Model is bounded above by $c_u \kappa^{1/3}$as$\kappa\downarrow0$ under mild regularity conditions. This bound of the same order of the previously proven lower bound, $\lambda(\kappa) \ge c_l \kappa^{1/3}$.