Paper detail

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

In this paper we identify the asymptotic tail of the distribution of the exit time $τ_C$ from a cone $C$ of an isotropic $α$-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 $Θ_t$. Under some additional regularity assumptions, the angular process $Θ_t$ 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 Lévy process related with Lamperti representation of the radial process, we prove that $$\mathbb{P}_x(τ_C>t)\sim h(x)t^{-κ_1}$$ as $t\rightarrow\infty$ for $h$ and $κ_1$ identified explicitly. The result extends the work of DeBlassie (1988) and Bañuelos and Smits (1997) concerning the Brownian motion.