Anomalous diffusion index for Lévy motions (Q2506287)

Let \(X=(X(t))_{t\in[0,\infty)}\) be a real-valued symmetric stochastic process. The authors define \[ \gamma_ X^ 0=\sup\{\gamma>0\colon \liminf_ {t\to\infty}\frac 1t\mathbb E| X(t)| ^{1/\gamma}>0\}, \] and they define the diffusion index \[ \gamma_ X=\inf\{\gamma\in(0,\gamma_ X^ 0]\colon \limsup_ {t\to\infty}\frac 1t\mathbb E| X(t)| ^{1/\gamma}<\infty\}. \] Various elementary properties of \(\gamma_ X\) and \(\gamma_ X^ 0\) are derived, and their relations to the diffusion coefficient \(\sigma_ X(x,t)=\lim_ {h\downarrow 0}\frac 1h \mathbb E[(X(t+h)-X(t))^ 2\mid X(t)=x]\) and to the diffusion constant \(D_ X=\lim_{t\to\infty}\frac 1{2t}\mathbb E X(t)^ 2 \) are examined. All these quantities are identified for a symmetric Lévy stable process, and special attention is payed to the special case of Brownian motion. Also relations to regularity properties of the distribution functions of the process are found. Among the explicit examples considered are the Ornstein-Uhlenbeck process and fractional Brownian motion.