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. (WINE20), 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 Saritac{c} (EC20), and shows that this prevalent bound in online algorithms is not optimal for this problem.
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