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