Abstract
In [6, Théorème VI. 1], it is shown that almost all sample paths of a given stable process (Zt)\(_{t \in [0,1]} \) of index \(\) belong to the Besov spaces \(B_{p,\infty }^{1/\alpha } \) with 1 ≤ p < α. Our aim is to establish a similar result for general Lévy processes (Xt)t ≥ 0. It will turn out that if we restrict the paths to compact time intervals (and put them zero elsewhere) then they belong to Besov spaces \(B_{p,\infty }^s \) for a certain choice of parameters s and p. Finally we extend the results obtained for Lévy processes to Markov processes, which are – in a certain sense – comparable with the given Lévy process.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aronson, D. G.: ‘Bounds for the fundamental solution of a parabolic equation’, Bull. Amer.Math. Soc. 73(1967), 890-896.
Berg, C. and Forst, G.: ‘Potential Theory on Locally Compact Abelian Groups’, Ergebnisse der Mathematik und ihrer Grenzgebiete, II. Ser. vol. 87. Springer-Verlag, Berlin, 1975.
Blumenthal, R.M. and Getoor, R. K.: ‘Sample Functions of Stochastic Processes with Stationary Independent Increments’, J. Math. Mech. 10(1961), 493-516.
Bochner, S.: Harmonic Analysis and the Theory of Probability.University of California Press, California Monographs in Math. Sci., Berkeley, 1955.
Ciesielski, Z.: ‘On Haar Functions and on the Schauder Basis of the Space C〈0, 1〉, Bull. Acad. Polon. Sci. 7(1959), 227-232.
Ciesielski, Z. Kerkyacharian, G. and Roynette, B.: ‘Quelques Espaces Fonctionnels Associes a des Processus Gaussiens’, Studia Math. 107.2(1993), 171-204.
Frazier, M. and Jawerth, B.: ‘Decomposition of Besov Spaces’, Indiana Univ.Math. J. 34(1985), 777-799.
Herren, V.: Lévy-Prozesse und Besov-Räume. Diplomarbeit, Universität Erlangen-Nürnberg, 1994.
Jacob, N. and Schilling, R. L.: ‘Subordination in the sense of S. Bochner - An approach through pseudo differential operators’, Math. Nachrichten 178(1996), 199-231.
Millar, P. W.: ‘Path Behavior of Processes with Stationary Independent Increments’, Z. Wahrscheinlichkeitstheorie verw. Geb., 17(1971), 53-73.
Protter, P.: Stochastic Integration and Differential Equations: A New Approach.Springer-Verlag, Berlin, 1990.
Schilling, R. L.: Zum Pfadverhalten von Markovschen Prozessen, die mit Lévy-Prozessen vergleichbar sind. Dissertation, Universität Erlangen-Nürnberg, 1994.
Schilling, R. L.: ‘Comparable processes and the Hausdorff dimension of their sample paths’. Stochastics and Stochastic Reports 57(1996), 89-110.
Triebel, H.: Theory of Function Spaces.Birkhäuser, Basel, 1983.
Triebel, H.: Theory of Function Spacess II. Birkhäuser, Basel, 1992.
Rights and permissions
About this article
Cite this article
Herren, V. Lévy-type Processes and Besov Spaces. Potential Analysis 7, 689–704 (1997). https://doi.org/10.1023/A:1017944015052
Issue Date:
DOI: https://doi.org/10.1023/A:1017944015052