Structure theorems for generalized random processes

Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on {M _p } on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on {M _p } and L ^r (R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y′(ω; t) = f(ω; t) is considered, where y is a generalized random process having a point value at t = 0 in the sense of Lojasiewicz. This paper was supported by the project Functional analysis, ODEs and PDEs with singularities, No. 144016, financed by the Ministry of Science, Republic of Serbia.