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Let eY he be the in®nitesimal generator of a Feller semigroup such that g I R n & he and ejg I R n is a pseudo-differential operator with symbol ÀpxY n satisfying kpY nk I 1 knk 2 and jIm pxY nj 0 Re pxY n. We show that the associated Feller process f t g t!0 on R n is a semimartingale, even a homogeneous diusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t 3 0 and I. To this end, we introduce various indices, e.g., b x I X inffk b 0 X lim knk3I sup kxÀyk 2aknk jpyY njaknk k 0g or d x I X inffk b 0 X lim inf knk3I inf kxÀyk 2aknk sup kk 1 jpyY knkjaknk k 0g, and obtain a.s. P x that lim t30 t À1ak sup s t k s À xk 0 or I according to k b b x I or k`d x I . Similar statements hold for the limit inferior and superior, and also for t 3 I. Our results extend the constant-coecient (i.e., Le vy) case considered by W. Pruitt [27].Mathematics Subject Classi®cation (1991): 60F15, 60J75, 60G17, 35S99, 60J35 Probab. Theory Relat. Fields 112, 565 ± 611 (1998) *Part of this work was done while the author was visiting the Universite d'Evry±Val d'Essonne, France. He would like to thank the sta of the maths department and, in particular, Prof. F. Hirsch, for their support and the good working conditions. Financial support by DFG post-doctoral fellowship Schi 419/1-1 is gratefully acknowledged.

p 20, Problem 3.5 be subsets of X be nonempty subsets of X p 20, Problem 3.5 (i) no proper subset B A no proper subset ∅ = B A p 20, Problem 3.9 Is this still true for the family B := {Br(x) : x ∈ Q n , r ∈ Q + } ? Denote by Br(x) an open ball in R n with centre x and radius r. Show that the Borel sets B(R n) are generated by the family of open balls B := {Br(x) : x ∈ R n , r > 0}. Is this still true for the family B := {Br(x) : x ∈ Q n , r ∈ Q + } ? p 26, line 6 below from Example 2.3(v) from Example 3.3(v) p 31, line 10 below

We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for Lévy processes and isotropic Lévy processes. Under some mild conditions on the characteristic exponent we calculate the asymptotic behaviour of the transition density as t → 0 and t → ∞ and show a ratio-limit theorem.MSC 2010: Primary: 60G51. Secondary: 60E10, 60F99, 60J35.Abstract. We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for Lévy processes and isotropic Lévy processes. Under some mild conditions on the characteristic exponent we calculate the asymptotic behaviour of the transition density as t → 0 and t → ∞ and show a ratio-limit theorem.MSC 2010: Primary: 60G51. Secondary: 60E10, 60F99, 60J35.

We derive explicitly the coupling property for the transition semigroup of a Lévy process and gradient estimates for the associated semigroup of transition operators. This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively. Our results can be applied to a large class of Lévy processes, including stable Lévy processes, layered stable processes, tempered stable processes and relativistic stable processes.

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Let (Z t ) t 0 be an R n -valued Lévy process. We consider stochastic differential equations of the formwhere Φ : R d → R d×n is Lipschitz continuous. We show that the infinitesimal generator of the solution process (X x t ) t 0 is a pseudo-differential operator whose symbol p :For a large class of Feller processes many properties of the sample paths can be derived by analysing the symbol. It turns out that the process (X x t ) t 0 is a Feller process if Φ is bounded and that the symbol is of the form p(x, ξ) = ψ(Φ ⊤ (x)ξ), where ψ is the characteristic exponent of the driving Lévy process. MSC 2010: 60J75; 47G30; 60H20; 60J25; 60G51; 60G17.