Fractional Operators and Fractionally Integrated Random Fields on Z

We consider fractional integral operators (I-T)ᵈ, d ∈ (-1,1) acting on functions g: Zν → R, ν ≥ 1, where T is the transition operator of a random walk on Zν. We obtain sufficient and necessary conditions for the existence, invertibility and square summability of kernels τ (s; d), s ∈ Zν of (I-T)ᵈ. Asymptotic behavior of τ (s; d) as |s| → ∞ is identified following local limit theorem for random walk. A class of fractionally integrated random fields X on Zν solving the difference equation (I-T)ᵈ X = ε with white noise on the right-hand side is discussed, and their scaling limits. Several examples including fractional lattice Laplace and heat operators are studied in detail.