Abstract This article describes a class of invariant Markov processes on certain totally disconnected groups. An invariant pseudodifferential operator on these groups, similar to the Vladimirov operator on the p -adic line, allows us to state an L 2 -abstract Cauchy problem for a homogeneous heat-type pseudodifferential equation. The fundamental solutions of these parabolic-type pseudodifferential equations give transition functions of time and space homogeneous Markov processes on these groups. Particularly interesting examples are polyadic rings, such as the ring of m -adic numbers, and the ring of finite adeles of the rational numbers.

中文翻译：

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某些完全不连通群上的伪微分算子和马尔可夫过程

摘要 本文描述了在某些完全断开的群上的一类不变马尔可夫过程。这些群上的不变伪微分算子，类似于 p-adic 线上的 Vladimirov 算子，允许我们陈述齐次热型伪微分方程的 L 2 -抽象柯西问题。这些抛物线型伪微分方程的基本解给出了这些群上时空齐次马尔可夫过程的转移函数。特别有趣的例子是多元环，例如 m-adic 数环和有理数的有限元环。