Fractional Operators and Fractionally Integrated Random Fields on <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">Z</mi></mrow></semantics></math></inline-formula><i>^ν</i>

Abstract

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 the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=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.

Keywords

• fractional differentiation/integration operators
• tempered fractional operators
• fractional random field
• random walk
• limit theorems
• long-range dependence