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MiS Preprint
42/1997
Function spaces as path spaces of Feller processes
Rene Schilling
Abstract
Let $\{X_i\}_i \geq 0$ be a Feller process taking values in $\mathbb{R}^x$ and with infinitesimal generator (A,D(A)). If the test functions are contained in D(A), $-A|C^\infty_c (\mathbb{R}^x)$ is a pseudo-differential operator p(x,D) with symbol $p(x,\varepsilon)$. We investigate local and global regularity properties of the sample paths $t \mapsto X_i$ in terms of (weighted) Besov $B^n_{pq} (\mathbb{R},p)$ and Triebel-Lizorkin $F^n_{pq} (\mathbb{R},p)$ spaces. The parameters for these spaces are determined by certain indices that describe the asymptotic behaviour of the symbol $p(x,\varepsilon)$. Our results improve previous papers of Ciesielski et al. (Studia Math.107, pp. 171-204), Roynette (Stochastics and Stochastics Reports43, pp. 221-260), and Herren (Potential Analysis7, pp. 689-704) on Lévy processes and Schilling (Math. Ann., to appear) on general Feller processes.
