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

TL;DR

This paper derives the precise asymptotic behavior of the tail distribution of the exit time from a cone for a class of isotropic self-similar Markov processes with a skew-product structure, extending previous Brownian motion results.

Contribution

It provides explicit asymptotics for the exit time distribution of a broad class of self-similar Markov processes, generalizing earlier Brownian motion findings.

Findings

Asymptotic tail probability behaves like a power law with explicit exponent.

Transition density of the angular process expressed via eigenfunctions.

Explicit formulas for the asymptotic decay rate and leading coefficient.

Abstract

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 $\Theta_t$. Under some additional regularity assumptions, the angular process $\Theta_t$ 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 $t\rightarrow\infty$ 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.