On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions

TL;DR

This paper explores the structure of reproducing kernel Hilbert spaces linked to fractional and bi-fractional Brownian motions, introducing a new complex-variable function that generalizes the Gamma function for these kernels.

Contribution

It provides novel decompositions of positive kernels and defines a new complex-variable function to analyze the associated Hilbert spaces.

Findings

Decomposition of kernels into sums or integrals of positive kernels

Introduction of a generalized Gamma function for complex variables

Insights into the structure of RKHS for fractional processes

Abstract

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider