On the exact asymptotics of exit time from a cone of an isotropic $\alpha$-self-similar Markov process with a skew-product structure

In this paper we identify the asymptotic tail of the distribution of the exit time ${\tau }_C$ from a cone $C$ of an isotropic $\alpha$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component ${\mathrm{\Theta }}_t$. Under some additional regularity assumptions, the angular process ${\mathrm{\Theta }}_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed L'evy process related with Lamperti representation of the radial process, we prove that $${\mathbb{P}}_x({\tau }_C>t)\sim h(x)t^{-{\kappa }_1}$$ as $t\to \mathrm{\infty }$ for $h$ and ${\kappa }_1$ identified explicitly. The result extends the work of DeBlassie (1988) and Ba~nuelos and Smits (1997) concerning the Brownian motion.