Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions (Q1868445)

The article is devoted to the small ball asymptotics for fractional Brownian motion in \(L_2\) norm. To get the asymptotic of the value \(\log (P\{\| B_H\| _2^2\leq \varepsilon \})\) the author considers the logarithmic moment generating function \[ \log (E\{\exp (-\lambda \| B_H\| _2^2 \}))=\log \Biggl( \biggl(\prod_{i=1}^\infty(1+2\lambda \lambda _i\biggr)\Biggr)^{-1/2}. \] Here \(\lambda _i\) are the eigenvalues of the covariance operator \(A.\) The author studies \(A\) by using its representation in the trigonometric basis in \(L_2\) and comparing the diagonal and off-diagonal terms. The key is the estimation of the eigenvalues of the sum of two self-adjoint compact operators. The precise asymptotics of \(\lambda _i\) allows to detect the asymptotics of the logarithmic moment generating function. This approach is different form the article [\textit{W. V. Li} and \textit{W. Linde}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 11, 1329--1334 (1998; Zbl 0922.60039)], where similar results were obtained.