On a diffusion on finite adeles and the Feynman-Kac integral

Let K be an algebraic number field. With K, we associate the ring of finite adeles [Formula: see text]. Following a recent result of Weisbart on diffusions on finite rational adeles [Formula: see text], we define the Vladimirov operator [Formula: see text] on [Formula: see text] and define the Brownian motion on the group [Formula: see text]. We also consider the Schrödinger operator [Formula: see text] with a potential operator V given by a non-negative continuous function v on [Formula: see text]. We prove a version of the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text].