Stochastic Mutation Semigroups: From Deterministic Collapse to Probabilistic Evolutionary Dynamics

The deterministic framework of mutation semigroups provides algebraic conditions for evolutionary collapse, but real mutation processes are inherently stochastic. This paper develops a comprehensive theory of stochastic mutation semigroups, where elementary mutations occur with empirically measured probabilities. A stochastic mutation semigroup is introduced as a Markov chain on the transformation semigroup generated by elementary mutation operators. We prove a stochastic transitivity threshold theorem under a positivity condition on the probability of rank reduction, correcting a logical gap in previous formulations, and we characterize collapse via the spectral properties of the associated Markov chain. Using HIV-1 sequence data from public databases and empirical mutation rates reported in the literature, the framework is illustrated through conceptual examples. Complete pseudocode is provided for the probabilistic pair-graph algorithm, along with convergence criteria and numerical examples demonstrating performance. The framework is further extended to infinite state spaces via topological semigroup theory and to time-varying mutation rates through dynamic L∗-classes. These results bridge the gap between abstract semigroup theory and evolutionary biology, offering a theoretical foundation for understanding stochastic mutation dynamics.