Extension theorem for rough paths via fractional calculus (Q2407547)

Studies in fractional calculus or fractional differintegrals of arbitrary orders, as that is also known, is now not a distant approach and in the past we have witnessed several of its applications in different fields of theoretical physics and within mathematics. Among such approaches, several of them are proposed for the study of the theory of rough paths, one of which is based on the use of fractional calculus. Those were introduced and extended by several authors, some of which are referred to in the present paper. In contrast to rough integration investigated and employed by \textit{T. J. Lyons} [Rev. Mat. Iberoam. 14, No. 2, 215--310 (1998; Zbl 0923.34056)] as the limit of a type of Riemann sums, the approach through fractional calculus has an advantage over the former in the sense that integrals along rough paths are described explicitly by ordinary Lebesgue integrals. Having spelled a self-contained introduction on the concepts of fractional calculus and integrals of weakly controlled path, which is a generalization of integrals in the context of rough paths, the Lyon's extension theorem is described in Section 3. Proofs of some results, in the form of theorems, are accommodated in Section 4, where as an application, an alternative proof of Lyon's extension theorem for geometric Hölder rough paths together with an explicit form of the extension map is given.