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

Very Strong Disorder for the Parabolic Anderson model in low dimensions