Using the geometric properties of Sobolev spaces of integer order and a duality condition, the covariance operators of a generalized random field and its dual can be factorized. Via this covariance factorization, a representation of the generalized random field is obtained as a stochastic equation driven by generalized white noise. This stochastic equation becomes a differential equation under the orthogonality of the dual random field. The solution to this equation satisfies the weak-sense Markov property of integer order. Furthermore, such a solution admits a mean-square series expansion in terms of the eigenfunctions associated with the pure point spectrum of the corresponding covariance operator. From this representation, the relationship between the covariance function and the Green function, associated with the corresponding deterministic problem, in the ordinary case, is derived under suitable conditions.