Monatsh Math (2018) 186:609–633 https://doi.org/10.1007/s00605-017-1109-z Probabilistic properties of generalized stochastic processes in algebras of generalized functions Snežana Gordi´ c 1 · Michael Oberguggenberger 2 · Stevan Pilipovi´ c 3 · Dora Seleši 3 Received: 10 April 2017 / Accepted: 7 October 2017 / Published online: 20 October 2017 © Springer-Verlag GmbH Austria 2017 Abstract Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established, which uniquely characterize the process, and relying on this result we prove the measurability of the corresponding random variables with values in the Colombeau algebra of compactly supported generalized constants endowed with the topology generated by sharp open balls. The generalized characteristic function and the generalized correlation function of Colombeau stochastic processes are introduced and their properties are investi- gated. It is shown that the characteristic function of classical stochastic processes can be embedded into the space of generalized characteristic functions. The generalized expectation and the generalized correlation function can be retrieved from the gener- alized characteristic function. The structural representation of the correlation function Communicated by A. Constantin. B Snežana Gordi´ c snezana.gordic@dmi.uns.ac.rs Michael Oberguggenberger michael.oberguggenberger@uibk.ac.at Stevan Pilipovi´ c pilipovic@dmi.uns.ac.rs Dora Seleši dora@dmi.uns.ac.rs 1 Faculty of Education, University of Novi Sad, Podgoriˇ cka 4, 25000 Sombor, Serbia 2 Institute for Basic Science in Engineering Sciences, University of Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria 3 Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovi´ ca 4, 21 000 Novi Sad, Serbia 123