  This paper provides information about the asymptotic behavior of a
one-dimensional Brownian polymer in random medium represented by a Gaussian
field $W$ on ${\mathbb{R}}_+\times{\mathbb{R}}$ which is white noise in time
and function-valued in space. According to the behavior of the spatial
covariance of $W$, we give a lower bound on the power growth (wandering
exponent) of the polymer when the time parameter goes to infinity: the polymer
is proved to be superdiffusive, with a wandering exponent exceeding any
$\alpha<3/5$.
