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 AK. Following a recent result of Weisbart on diffusions on finite rational adeles AQ, we define the Vladimirov operator delta(AK) on A(K) and define the Brownian motion on the group AK. We also consider the Schrodinger operator -H-AK = -delta(AK) + V with a potential operator V given by a non-negative continuous function v on A(K). We prove a version of the Feynman-Kac formula for the Schrodinger semigroup generated by -HAK. Published under an exclusive license by AIP Publishing.