Growth and Hölder conditions for the sample paths of Feller processes

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)| ≤ c0 Re p(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∞ := inf{λ > 0 : lim∥ξ∥→∞ sup∥x-y∥≤2/∥ξ∥|p(y, ξ)|/∥ξ∥λ = 0} or δx∞ := inf{λ > 0 : lim inf∥ξ∥→∞ inf∥x-y∥≤2/∥ξ∥ sup∥ε∥≤1 |p(y, ∥ξ∥ε)|/∥ξ∥λ = 0}, and obtain a.s. (ℙx) that limt→0 t-1/λ sups≤t ∥Xs - x∥ = 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].