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 $\psi(\lambda)-\frac{\lambda}{2}\psi'(\lambda)$ varies regularly, as $\frac{t\psi(r^{-1})^2}{\psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})} \to 0$ the tail probability $\mathbb{P}(|X_t|\geq r)$ is asymptotically equal to a constant times $ t( \psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})).$