Growth and Hölder conditions for the sample paths of Feller processes

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., 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→0t−1/λs≤t|Xs−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].