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Let us consider a solution of a one-dimensional stochastic differential equation driven by a standard Brownian motion with time-inhomogeneous drift coefficient ${\rho sgn\left(x\right)\left|x\right|}^{\alpha }/t^{\beta }$. This process can be viewed as a Brownian motion evolving in a potential, possibly singular, depending on time. We prove results on the existence and uniqueness of solution, 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...