Schilling, Rene (1998) Growth and Hölder conditions for the sample paths of Feller processes. Probability Theory and Related Fields, 112 (4). pp. 565-611. ISSN 0178-8051

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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].

Item Type:	Article
Schools and Departments:	School of Mathematical and Physical Sciences > Mathematics
Depositing User:	EPrints Services
Date Deposited:	06 Feb 2012 19:50
Last Modified:	09 Jul 2012 15:35
URI:	http://sro.sussex.ac.uk/id/eprint/22444

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