Last exit times for transient semistable processes

Abstract

Let L B a be the last exit time from the ball B a = { | x | < a } for a nondegenerate transient α-semistable process { X t } on R d . The problem to determine the set T defined by T = { 0 } ∪ { η > 0 : E [ L B a η ] < ∞ } is studied. The process { X t } is called first-class or second-class according as it is strictly α-semistable or not. A unique location parameter τ ∈ R d is introduced in connection to the space–time relation of { X t } ; τ = 0 if and only if { X t } is first-class; τ is the drift if 0 < α < 1 and the center if 1 < α ⩽ 2 . The set T is determined in the case d = 1 and in the following cases with d ⩾ 2 : (i) 0 < α < 1 ; (ii) 1 ⩽ α ⩽ 2 and τ = 0 ; (iii) 1 ⩽ α < 2 , τ ≠ 0 , and σ ( { − τ / | τ | } ) > 0 ; (iv) 1 ⩽ α < 2 , τ ≠ 0 , and − τ / | τ | ∉ C σ . Here σ is the spherical component of the Lévy measure, and C σ is a set defined by the support of σ. Weak transience and strong transience correspond to 1 ∉ T and 1 ∈ T , respectively, and they are completely classified in terms of d, α, τ, and another parameter β. Applications to the Spitzer type limit theorems involving capacity are given.