Asymptotical properties of distributions of isotropic L\' evy processes

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic L'evy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1. The asymptotic expressions are given in terms of the radial part of characteristic exponent $\psi$ and its derivative. In particular, when $\psi (\lambda )-\frac{\lambda }{2}{\psi }^{\mathrm{\prime }}(\lambda )$ varies regularly, as $\frac{t\psi (r^{-1}{)}^2}{\psi (r^{-1})-(2r{)}^{-1}{\psi }^{\mathrm{\prime }}(r^{-1})}\to 0$ the tail probability $\mathbb{P}(\mathrm{\mid }{X}_t\mathrm{\mid }\ge r)$ is asymptotically equal to a constant times $t(\psi (r^{-1})-(2r{)}^{-1}{\psi }^{\mathrm{\prime }}(r^{-1}))\mathrm{.}$