LAUSR

In this paper, we establish the precise asymptotics 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 asymptotics of the tail probability of subordinators when the scaling order is between 0 and 1 including 1. The asymptotics 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 $\P(|X_t|\geq r)$ is asymptotically equal to a constant times $ t( \psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})).$