ArticleOriginal scientific text
Title
Régularité du temps local brownien dans les espaces de Besov-Orlicz
Authors 1
Affiliations
- Institut E. Cartan, B.P., 239 54506 Vandœ uvre-lès-Nancy Cedex, France
Abstract
Let be a linear Brownian motion and (L(t,x), t > 0, x ∈ ℝ) its local time. We prove that for all t > 0, the process (L(t,x), x ∈ [0,1]) belongs almost surely to the Besov-Orlicz space with .
Bibliography
- [BY] M. T. Barlow and M. Yor, Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times, J. Funct. Anal. 49 (1982), 198-229.
- [Be1] S. M. Berman, Sojourns and Extremes of Stochastic Processes, Wadsworth & Brooks/Cole, Pacific Grove, 1992.
- [Be2] S. M. Berman, Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc. 137 (1969), 277-299.
- [Bo] A. N. Borodin, Distribution of integral functionals of Brownian motion, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 119 (1982), 19-38 (in Russian).
- [BR] B. Boufoussi et B. Roynette, Le temps local brownien appartient p.s. à l'espace de Besov
, C. R. Acad. Sci. Paris Sér. I 316 (1993), 843-848. - [C1] Z. Ciesielski, On the isomorphisms of the spaces
and m, Bull. Acad. Polon. Sci. 8 (1960), 217-222. - [C2] Z. Ciesielski, Orlicz spaces, spline systems, and Brownian motion, Constr. Approx. 9 (1993), 191-222.
- [CKR] Z. Ciesielski, G. Kerkyacharian et B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), 171-204.
- [K1] F. B. Knight, Random walks and a sojourn density process of Brownian motion, Trans. Amer. Math. Soc. 109 (1963), 56-86.
- [K2] F. B. Knight, Essentials of Brownian Motion and Diffusion, Math. Surveys 18, Amer. Math. Soc., Providence, 1981.
- [KR] M. A. Krasnosel'skiĭ and Ya. B. Rutitskiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
- [L] P. Lévy, Le Mouvement Brownien, Mém. Sci. Math. 126, Gauthier-Villars, Paris, 1954.
- [Lu] W. A. J. Luxemburg, Banach function spaces, Thesis, Technische Hogeschool te Delft, 1955; MR 17 (1956), 285.
- [M] H. P. McKean, A Hölder condition for Brownian local time, J. Math. Kyoto Univ. 1 (1962), 195-201.
- [P] E. Perkins, Local time is a semi-martingale, Z. Warsch. Verw. Gebiete 60 (1982), 79-117.
- [Ra] D. B. Ray, Sojourn times of diffusion processes, Illinois J. Math. 7 (1963), 615-630.
- [RY] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1991.
- [Ro] B. Roynette, Mouvement brownien et espaces de Besov, Stochastics Stochastics Rep. 43 (1993), 221-260.
- [T] H. Trotter, A property of Brownian motion paths, Illinois J. Math. 2 (1958), 425-433.
- [W] N. Wiener, Generalized harmonic analysis, Acta Math. 55 (130), 117-258.