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In this paper we identify the asymptotic tail of the distribution of the exittime $\tau_C$ from a cone $C$ of an isotropic $\alpha$-self-similar Markovprocess $X_t$ with a skew-product structure, that is $X_t$ is a product of itsradial process and independent time changed angular component $\Theta_t$. Undersome additional regularity assumptions, the angular process $\Theta_t$ killedon exiting from the cone $C$ has the transition density that could be expressedin terms of a complete set of orthogonal eigenfunctions with correspondingeigenvalues of an appropriate generator. Using this fact and some asymptoticproperties of the exponential functional of a killed L\'evy process relatedwith Lamperti representation of the radial process, we prove that$$\mathbb{P}_x(\tau_C>t)\sim h(x)t^{-\kappa_1}$$ as $t\rightarrow\infty$ for$h$ and $\kappa_1$ identified explicitly. The result extends the work ofDeBlassie (1988) and Ba\~nuelos and Smits (1997) concerning the Brownianmotion.

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