Asymptotic mass distribution speed for the one-dimensional heat equation with constant drift and stationary potential

We study the long-time behavior of the solution u(t,x) of a Cauchy problem for the one-dimensional heat equation with constant drift and random potential in the quenched setting: ut=12uxx+hux+ξu. The initial function is compactly supported. For bounded stationary ergodic potential ξ, we show that u is asymptotically (t→∞) concentrated in a ball of radius o(t) and center vht which is independent of the realization of the random potential. There is a critical drift value hcr where we observe a change from sublinear (vh=0) to linear (0<vh⩽h) mass propagation.