  This paper is concerned with two related types of directed polymers in a
random medium. The first one is a d-dimensional Brownian motion living in a
random environment which is Brownian in time and homogeneous in space. The
second is a continuous-time random walk on the lattice Z^d, in a random
environment with similar properties as in continuous space. The case of a
space-time white noise environment can be acheived in this second setting. By
means of some Gaussian tools, we estimate the free energy of these models at
low temperature, and give some further information on the strong disorder
regime of the objects under consideration.
