A Sufficient Condition for Two Markov Semigroups to Commute

Abstract

Let $\{P^{(k)}_t, t \geq 0\}, k = 1, 2$, be two Markov semigroups on $\hat{C}(E)$, the space of continuous functions on a separable, locally compact metric space $E$ which tend to zero at infinity. In this article, we derive a sufficient condition for the two semigroups to commute, in the sense that for each $s \geq 0, t \geq 0$ and each $f \in \hat{C}(E), P^{(1)}_s P^{(2)}_t f = P^{(2)}_t P^{(1)}_s f$.