A model of continuous time polymer on the lattice

In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice ${\mathbb{Z}}^d$, and the environment is a collection $\{W(t,x);t\ge 0,x\in {\mathbb{Z}}^d\}$ of i.i.d. Brownian motions.