A single seller faces a sequence of buyers with unit demand. The buyers are forward‐looking and long‐lived. Each buyer has private information about his arrival time and valuation where the latter evolves according to a geometric Brownian motion. Any incentive‐compatible mechanism has to induce truth‐telling about the arrival time and the evolution of the valuation. We establish that the optimal stationary allocation policy can be implemented by a simple posted price. The truth‐telling constraint regarding the arrival time can be represented as an optimal stopping problem that determines the first time at which the buyer participates in the mechanism. The optimal mechanism thus induces progressive participation by each buyer: he either participates immediately or at a future random time.