Localization for a Random Walk in Slowly Decreasing Random Potential

We consider a continuous time random walk X in a random environment on ?⁺ such that its potential can be approximated by the function V:?⁺→? given by $V(x)=sigma W(x) -frac {b}{1-alpha}x^{1-alpha}$ where σW a Brownian motion with diffusion coefficient σ>0 and parameters b, α are such that b>0 and 0<α<1/2. We show that P-a.s. (where P is the averaged law) $lim_{ttoinfty} frac{X_{t}}{(C^{*}(lnln t)^{-1}ln t)^{frac{1}{alpha}}}=1$ with $C^{*}=frac{2alpha b}{sigma^{2}(1-2alpha)}$ . In fact, we prove that by showing that there is a trap located around $(C^{*}(lnln t)^{-1}ln t)^{frac{1}{alpha}}$ (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 ln² t.