Generalized two-parameter Lebesgue-Stieltjes integrals and their applications to fractional Brownian fields

Abstract

We consider two-parameter fractional integrals and Weyl, Liouville, and Marchaut derivatives and substantiate some of their properties. We introduce the notion of generalized two-parameter Lebesgue-Stieltjes integral and present its properties and computational formulas for the case of differentiable functions. The main properties of two-parameter fractional integrals and derivatives of Hölder functions are considered. As a separate case, we study generalized two-parameter Lebesgue-Stieltjes integrals for an integrator of bounded variation. We prove that, for Hölder functions, the integrals indicated can be calculated as the limits of integral sums. As an example, generalized two-parameter integrals of fractional Brownian fields are considered.