A general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes. "Stochastically positive KMS systems," "periodic OS-positive stochastic processes," "periodic positive semigroup structures" are defined and shown to be equivalent. The objects of the Tomita-Takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven. A (two-sided) Markov property for the stochastic process is related to a cyclicity property of the KMS state. Perturbations are constructed by the Feynman-Kac-Nelson formula. The general framework is applied to KMS states given by density matrices. © 1981.
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