Asymptotic formulas for symmetric stable semigroups (Q1342743)

The author gives a new and elementary proof of the fact that the transition density \(\mu^ \alpha_ t (x)\) of a symmetric \(\alpha\)- stable \((0 < \alpha \leq 2)\) semigroup is asymptotically \(c_{n, \alpha} t(x^ 2 + t^{2/ \alpha})^{-(n + \alpha)/2}\) as \(x^ 2 t^{-2/ \alpha} \to \infty\). The constant \(c_{n, \alpha}\) is explicitly given. The proof is based on the fact that \(\alpha\)-stable semigroups are subordinate (in the sense of Bochner) to a Brownian semigroup.