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Growth and Hölder conditions for the sample paths of Feller processes
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  • Published: December 1998

Growth and Hölder conditions for the sample paths of Feller processes

  • René L. Schilling1 

Probability Theory and Related Fields volume 112, pages 565–611 (1998)Cite this article

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Abstract.

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C c ∞(ℝn)⊂D(A) and A|C c ∞(ℝn) 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 ℝn 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. (ℙx) 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].

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Authors and Affiliations

  1. The Nottingham Trent University, Mathematics Department, Burton Street, Nottingham NG1 4BU, United Kingdom. e-mail: rls@maths.ntu.ac.uk, , , , , , GB

    René L. Schilling

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  1. René L. Schilling
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Received: 21 July 1997 / Revised version: 26 January 1998

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Schilling, R. Growth and Hölder conditions for the sample paths of Feller processes. Probab Theory Relat Fields 112, 565–611 (1998). https://doi.org/10.1007/s004400050201

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  • Issue Date: December 1998

  • DOI: https://doi.org/10.1007/s004400050201

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  • Mathematics Subject Classification (1991): 60F15
  • 60J75
  • 60G17
  • 35S99
  • 60J35
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