The Perron-Frobenius Theorem for Markov Semigroups

Abstract

Let $P_t^V$, $t\ge 0$, be the Schrodinger semigroup associated to a potential $V$ and Markov semigroup $P_t$, $t\ge 0$, on $C(X)$. Existence is established of a left eigenvector and right eigenvector corresponding to the spectral radius $e^{{\lambda }_{0}t}$ of $P_t^V$, simultaneously for all $t\ge 0$. This is derived with no compactness assumption on the semigroup operators.