Very strong disorder for the parabolic Anderson model in low dimensions

We study the free energy of the parabolic Anderson model, a time-continuous model of directed polymers in the random environment. We prove that in dimensions 1 and 2, the free energy is always negative, meaning that very strong disorder always holds. The result for discrete polymers in dimension 2, as well as better bounds on the free energy in dimension 1, was first obtained by Hubert Lacoin in Lacoin (2010), and the goal of this paper is to adapt his proof to the parabolic Anderson model.