Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an -generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when ∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.