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

Abstract

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