Function Spaces as Path Spaces of Feller Processes

Abstract

Let {Xt}t ≥ 0 be a Feller process with infinitesimal generator (A, D(A)). If the test functions are contained in D(A), —A |C∞c (ℝn) is a pseudo–differential operator p(x, D) withsymbol p(x, ξ). We investigate local and global regularity properties of the sample paths t ↦ Xt in terms of (weighted) Besov Bspq (ℝ, ρ) and Triebel–Lizorkin Fspq (ℝ, ρ) spaces. The parameters for these spaces are determined by certain indices that describe the asymptotic behaviour of the symbol p(x, ξ). Our results improve previous papers on Lévy [5, 9] and Feller processes [22].