Probabilistic properties of generalized stochastic processes in algebras of generalized functions

Abstract

© 2017, Springer-Verlag GmbH Austria. 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 investigated. 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 generalized characteristic function. The structural representation of the correlation function which is supported on the diagonal is given. Examples of generalized characteristic functions related to Gaussian Colombeau stochastic processes are given.