Asymptotics of Karhunen–Loève Eigenvalues for Sub-Fractional Brownian Motion and Its Application

Abstract

In the present paper, the Karhunen–Loève eigenvalues for a sub-fractional Brownian motion are considered. Rigorous large n asymptotics for those eigenvalues are shown, based on the functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotical estimates for the small L2-ball probabilities for a sub-fractional Brownian motion are derived. Asymptotic analysis on the Karhunen–Loève eigenvalues for the corresponding “derivative” process is also established.

New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators

“…Fractional differential systems and evolution systems have been studied extensively owing to its widespread backgrounds of some scientific and engineering realms, such as signal processing, finance, anomalous diffusion phenomena, heat conduction, etc. We refer readers to [1][2][3][4] for further detailed information. On the other side, controllability has gained a lot of importance and interest, and it plays a significant role in the description of various dynamical problems [5][6][7][8].…”