In this article a class of Markov processes on the ring of finite adèles of the rational numbers are introduced. A class of non-Archimedean metrics on $\mathbb{A}_{f}$Af are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on ${L^2}(\mathbb{A}_{f})$L2(Af).. The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on $\mathbb{A}_{f}$Af which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adèles.

中文翻译：

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Adèles 上的伪微分算子和马尔可夫过程

本文介绍了有理数有限元环上的一类马尔可夫过程。选择 $\mathbb{A}_{f}$Af 上的一类非阿基米德度量，以将这个环描述为一个一般的多元环，并在 ${L 上引入一系列伪微分算子和抛物线型方程^2}(\mathbb{A}_{f})$L2(Af).. 这些抛物线方程的基本解决定了 $\mathbb{A}_{f}$ 上时空齐次马尔可夫过程的转移函数Af 在乘以单位时不变。考虑到无限的地方 ℝ，我们将这些结果扩展到完整的 adèles 环。