Summary
In this paper, we determine Onsager-Machlup functionals for a variety of norms on Wiener space which includes among others Hölder norms for every 0<α<1/2, as well as Besov or Sobolev type norms. We basically require the knowledge of the small ball probabilities for the Wiener measure and use versions of the norms which are rotationaly invariant on the range of the Brownian paths, a property of crucial importance in our approach.
Article PDF
Similar content being viewed by others
References
Airault, H. and Malliavin, P., Intégration géométrique sur l'espace de Wiener, Bull. Sci. Math. 112, 1988, pp. 3–52
Baldi, P. and Roynette, B., Some exact equivalents for the Brownian motion in Hölder norm, Probab. Theory Relat. Fields 93, 1992, pp. 457–484.
Borell, C., “A note on Gauss measures which agree on small balls”, Annal. Inst. Henri Poincaré, Vol. XIII, no3, 1977, pp. 231–238
Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland, 1989, pp. 532–539
Ledoux, M., Isoperimetry and Gaussian analysis, Ecole d'Eté de Probabilités de Saint-Flour 1994, Lecture Notes in Math. Springer-Verlag, to appear
Shepp, L.A. and Zeitouni, O., “A note on conditional exponential moments and the Onsager Machlup functional”, Annals of Probability, 20 (1992), pp. 652–654
Shepp, L.A. and Zeitouni, O., “Exponential estimates for convex norms and some applications”, Barcelona Seminar on Stochastic Analysis, St Feliu de Guixols, 1991, Progress in Prob. vol. 32, Birkhäuser Verlag. Basel. Boston. Berlin, 1993. pp. 203–215
Stolz, W., “Une méthode élémentaire pour l'évaluation de petites boules browniennes”, C. R. Acad. Sci. Paris, t.316, Série I, pp. 1217–1220, 1993
Triebel, H., Theory of Function Spaces, Birkhäuser Verlag. Basel. Boston. Stuttgart, 1983
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Capitaine, M. Onsager-Machlup functional for some smooth norms on Wiener space. Probab. Th. Rel. Fields 102, 189–201 (1995). https://doi.org/10.1007/BF01213388
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01213388