Markov processes on the adeles and Dedekind's zeta function

Let $K$ be an algebraic number field. We construct an additive Markov process $X_t^{K_{\mathbb{A}}}$ on the ring of adeles $K_{\mathbb{A}},$ whose coordinates $X_t^{(v)}$ are independent and use this process to give a probabilistic interpretation of the Dedekind zeta function ${\zeta }_K(s),$ for $\text{re}s>1.$ This note extends a recent work of Yasuda [J. Theor. Probab. 23(3):748–769, 2010] where the case of the field $K=\text{Q}$ of rational numbers was considered.