Growth and Hölder conditions for the sample paths of Feller processes

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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