  We consider a continuous time random walk $X$ in random environment on $\Z^+$
such that its potential can be approximated by the function $V: \R^+\to \R$
given by $V(x)=\sig W(x) -\frac{b}{1-\alf}x^{1-\alf}$ where $\sig W$ a Brownian
motion with diffusion coefficient $\sig>0$ and parameters $b$, $\alf$ are such
that $b>0$ and $0<\alf<1/2$. We show that $\P$-a.s.\ (where $\P$ is the
averaged law) $\lim_{t\to \infty} \frac{X_t}{(C^*(\ln\ln t)^{-1}\ln
t)^{\frac{1}{\alf}}}=1$ with $C^*=\frac{2\alf b}{\sig^2(1-2\alf)}$. In fact, we
prove that by showing that there is a trap located around $(C^*(\ln\ln
t)^{-1}\ln t)^{\frac{1}{\alf}}$ (with corrections of smaller order) where the
particle typically stays up to time $t$. This is in sharp contrast to what
happens in the "pure" Sinai's regime, where the location of this trap is random
on the scale $\ln^2 t$.
