Probability Theory and Related Fields

, Volume 112, Issue 4, pp 565–611

Growth and Hölder conditions for the sample paths of Feller processes

  • René L. Schilling

DOI: 10.1007/s004400050201

Cite this article as:
Schilling, R. Probab Theory Relat Fields (1998) 112: 565. doi:10.1007/s004400050201


Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that Cc(ℝn)⊂D(A) and A|Cc(ℝn) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|c(1+|ξ|2) and |Imp(x,ξ)|≤c0Rep(x,ξ). We show that the associated Feller process {Xt}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., βx:={λ>0:lim|ξ|→∞|xy|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δx:={λ>0:liminf|ξ|→∞|xy|≤2/|ξ||ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙx) that lim t→0t−1/λst|Xsx|=0 or ∞ according to λ>βx or λ<δx. 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].

Mathematics Subject Classification (1991): 60F1560J7560G1735S9960J35

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • René L. Schilling
    • 1
  1. 1.The Nottingham Trent University, Mathematics Department, Burton Street, Nottingham NG1 4BU, United Kingdom. e-mail: