Quantum rolling tachyon

We consider the quantum treatment of the rolling tachyon background describing the decay of D-branes in the limit of weak string coupling. We focus on the propagation of an open string in the fluctuating background and show how the boundary string action is modified by quantum effects. A bilocal term in the boundary action is generated which, however, does not spoil the vanishing of the $\beta$ function at one loop. The propagation of an open string for large times is found to be very strongly suppressed.