  Let us consider a solution of the time-inhomogeneous stochastic differential
equation driven by a Brownian motion with drift coefficient $b(t,x)=\rho\,{\rm
sgn}(x)|x|^\alpha/t^\beta$. This process can be viewed as a distorted Brownian
motion in a potential, possibly singular, depending on time. After obtaining
results on existence and uniqueness of solution, we study its asymptotic
behaviour and made a precise description, in terms of parameters $\rho,\alpha$
and $\beta$, of the recurrence, transience and convergence. More precisely,
asymptotic distributions, iterated logarithm type laws and rates of transience
and explosion are proved for such processes.
