Key Fractional Gaussian Processes

This chapter explores the key fractional Gaussian processes, including fractional Brownian motion (fBm), its odd and even variants, and sub-fBm. It provides the essential tools and insights needed to understand the generalized fBm. The chapter demonstrates that both odd fBm (ofBm) and even fBm (efBM) can be uniquely characterized as the odd and even parts, respectively, of a two-sided fBm. It focuses on the core stochastic characteristics of ofBm and efBm. Beyond the variance of increments, integral representations provide a powerful tool for characterizing these processes. They often simplify proofs and reveal deeper mathematical structures. The chapter recalls the main results obtained by Dzhaparidze–van Zanten and their subsequent series representations. These representations are often expressed in terms of Bessel functions, utilizing their zeros to form an orthogonal basis for the relevant function spaces, offering a constructive approach to the fBm.