  We study a continuous and infinite time horizon counterpart to the classic
prophet inequality, which we term the stationary prophet inequality problem.
Here, copies of a good arrive and perish according to Poisson point processes.
Buyers arrive similarly and make take-it-or-leave-it offers for unsold items.
The objective is to maximize the (infinite) time average revenue of the seller.
  Our main results are pricing-based policies which (i) achieve a
$1/2$-approximation of the optimal offline policy, which is best possible, and
(ii) achieve a better than $(1-1/e)$-approximation of the optimal online
policy. Result (i) improves upon bounds implied by recent work of Collina et
al. (WINE'20), and is the first optimal prophet inequality for a stationary
problem. Result (ii) improves upon a $1-1/e$ bound implied by recent work of
Aouad and Sarita\c{c} (EC'20), and shows that this prevalent bound in online
algorithms is not optimal for this problem.
