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Contents of PMS, Vol. 41, Fasc. 1,
pages 25 - 38
DOI: 10.37190/0208-4147.41.1.3
Published online 12.3.2021
 

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

Zbigniew Palmowski
Longmin Wang

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 an independent time changed angular component \(\Theta_t\). Under some additional regularity assumptions, the angular process \(\Theta_t\) killed on exiting the cone \(C\) has a transition density that can 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 to the Lamperti representation of the radial process, we prove that \[P_x(\tau_C>t)\sim h(x)t^{-\kappa_1}\] as \(t\rightarrow\infty\) for \(h\) and \(\kappa_1\) identified explicitly. The result extends the work of De Blassie (1988) and Banuelos and Smits (1997) concerning the Brownian motion.

2010 AMS Mathematics Subject Classification: Primary 31B05; Secondary 60J45.

Keywords and phrases: \(\alpha\) -self-similar process, cone, exit time, skew-product structure, Lamperti representation, exponential functional, Brownian motion.

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