Monoid actions of trace monoids over finite sets are powerful models of concurrent systems---for instance they encompass the class of 1-safe Petri nets. We characterise Markov measures attached to concurrent systems by finitely many parameters with suitable normalisation conditions. These conditions involve polynomials related to the combinatorics of the monoid and of the monoid action. These parameters generalise to concurrent systems the coefficients of the transition matrix of a Markov chain. A natural problem is the existence of the uniform measure for every concurrent system. We prove this existence under an irreducibility condition. The uniform measure of a concurrent system is characterised by a real number, the characteristic root of the action, and a function of pairs of states, the Parry cocyle. A new combinatorial inversion formula allows to identify a polynomial of which the characteristic root is the smallest positive root. Examples based on simple combinatorial tilings are studied.

中文翻译：

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并发系统的马尔可夫动力学

跟踪幺半群在有限集上的幺半群动作是并发系统的强大模型——例如，它们包含 1-safe Petri 网类。我们通过具有合适归一化条件的有限多个参数来表征附加到并发系统的马尔可夫度量。这些条件涉及与幺半群和幺半群作用的组合相关的多项式。这些参数将马尔可夫链的转移矩阵的系数推广到并发系统。一个自然的问题是每个并发系统都存在统一的度量。我们在不可约条件下证明这种存在性。并发系统的统一测度由实数、动作的特征根和状态对的函数 Parry cocyle 表征。新的组合反演公式允许识别特征根是最小正根的多项式。研究了基于简单组合拼贴的示例。