  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 $\Z^d$, and
the environment is a collection $\{W(t,x);t\ge 0, x\in \Z^d\}$ of i.i.d.
Brownian motions.
