Markov processes on the adeles and representations of Euler products

In this article we construct Markov processes on the ring of adeles. Their transition probabilities are given as solutions of Kolmogorov's differential equations, and the resulting processes have independent p-components which are p-adic-valued additive processes investigated in the last two decades. Stochastic analysis on the p-adic fields has clarified some crucial differences of these processes from those on Euclidean spaces and has given new methods in functional analysis on local fields. On the other hand, when we deal with all prime numbers simultaneously and focus on analysis on the ring of adeles, we can peek into the real and complex fields through adelic formulae such as Euler product representations. We take up local and global exit times of adelic processes from some subsets and give some representation formulae relating them to Euler products.