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We consider quasi-free stochastically positive ground and thermal states onWeyl algebras in Euclidean time formulation. In particular, we obtain a newderivation of a general form of thermal quasi-free state and give conditionswhen such state is stochastically positive i.e. when it defines periodicstochastic process with respect to Euclidean time, so called thermal process.Then we show that thermal process completely determines modular structurecanonically associated with quasi-free state on Weyl algebra. We discuss avariety of examples connected with free field theories on globally hyperbolicstationary space-times and models of quantum statistical mechanics.Comment: 46 pages, amste

Stochastically positive structures on Weyl algebras. The case of quasi-free states