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Zbl 1086.60028
Sato, Ken-iti; Watanabe, Toshiro
Last exit times for transient semistable processes. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 5, 929-951 (2005). ISSN 0246-0203

Let $L_{B_a}$ be the last exit time from the ball $B_a=\{\vert x\vert < a \}$ for a nondegenerate transient $\alpha$-semistable process $\{X_t\}$ on ${\Bbb R}^d$. The problem to determine the set ${\cal T}$ defined by ${\cal T}=\{0\} \cup \{\eta > 0 : E[L^{\eta}_{B_a}] < \infty\}$ is studied. The process $\{X_t\}$ is called first-class or second-class according as it is strictly $\alpha$-semistable or not. A unique location parameter $\tau \in {\Bbb R}^d$ is introduced in connection to the space-time relation of $\{X_t\}$, $\tau=0$, if and only if $\{X_t\}$ is first-class; $\tau$ is the drift if $0 < \alpha < 1$ and the center if $1 < \alpha \le 2$. The set ${\cal T}$ is determined in the case $d=1$ and in the following cases with $d \ge 2$: (i) $0 < \alpha < 1$; (ii) $1 \le \alpha \le 2$ and $\tau = 0$; (iii) $1 \le \alpha < 2$, $\tau \neq 0$, and $\sigma(\{\tau/\vert \tau\vert \}) > 0$; (iv) $1 \le \alpha < 2$, $\tau \neq 0$, and $-\tau/\vert \tau\vert \notin C_{\sigma}$. Here $\sigma$ is the spherical component of the Lévy measure, and $C_{\sigma}$ is a set defined by the support of $\sigma$. Weak transience and strong transience correspond to $1 \notin {\cal T}$ and $1 \in {\cal T}$, respectively, and they are completely classified in terms of $d$, $\alpha$, $\tau$, and another parameter $\beta$. Applications to the Spitzer type limit theorems involving capacity are given.
[Pavel Gapeev (Moskva)]

MSC 2000:: *60G51 Processes with independent increments
60G52 Stable processes

Keywords: Lévy process; stable process

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