Complex-Order Scale-Invariant Operators and Self-Similar Processes

TL;DR

This paper extends fractional derivatives to complex orders using Fourier domain methods, introduces complex-valued self-similar processes, and analyzes their decay, regularity, and stationarity properties.

Contribution

It develops a novel framework for complex-order scale-invariant operators and constructs new self-similar processes with complex Hurst indices.

Findings

Operators exhibit specific decay properties

Processes are self-similar and stationary

Regularity analyzed in Sobolev spaces

Abstract

Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also belong to this family. In this paper, we extend the fractional operators to complex-order operators by constructing them in the Fourier domain. We analyze these operators in details with a special emphasis on the decay properties of the outputs. We further use these operators to introduce a family of complex-valued stable processes that are self-similar with complex-valued Hurst indices. These processes are expressed via the characteristic functionals over the Schwartz space of functions. Besides the self-similarity and stationarity, we study the regularity (in terms of Sobolev spaces) of the proposed processes.