, Volume 112, Issue 4, pp 565-611

Growth and Hölder conditions for the sample paths of Feller processes

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Abstract.

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C c (ℝ n )⊂D(A) and A|C c (ℝ n ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|c(1+|ξ|2) and |Imp(x,ξ)|≤c 0Rep(x,ξ). We show that the associated Feller process {X t } t ≥0 on ℝ n is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β x :={λ>0:lim |ξ|→∞ | x y |≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ x :={λ>0:liminf |ξ|→∞ | x y |≤2/|ξ| |ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ x ) that lim t →0 t −1/λ s t |X s x|=0 or ∞ according to λ>β x or λ<δ x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].

Received: 21 July 1997 / Revised version: 26 January 1998