  This paper provides information about the asymptotic behavior of a
one-dimensional Brownian polymer in random medium represented by a space-time
Gaussian field W assumed to be white noise in time and function-valued in
space. According to the behavior of the spatial covariance 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 $\alpha<3/5$.
