Onsager-Machlup Functional for Uniformly Elliptic Time-Inhomogeneous Diffusion

  • Koléhè A. Coulibaly-Pasquier
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


In this paper, we will compute the Onsager-Machlup functional of an inhomogeneous uniformly elliptic diffusion process. This functional is very similar to the corresponding functional for homogeneous diffusions; indeed, the only difference come from the infinitesimal variation of the volume. We will also use the Onsager-Machlup functional to study small ball probability for weighted sup-norm of some inhomogeneous diffusion.


  1. 1.
    M. Capitaine, On the Onsager-Machlup functional for elliptic diffusion processes, in Séminaire de Probabilités, XXXIV. Lecture Notes in Mathematics, vol. 1729 (Springer, Berlin, 2000), pp. 313–328Google Scholar
  2. 2.
    A.K. Coulibaly-Pasquier, Brownian motion with respect to time-changing riemannian metrics, applications to Ricci flow. Ann. Inst. Henri Poincaré Probab. Stat. 47(2), 515–538 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    A.K. Coulibaly-Pasquier, Some stochastic process without birth, linked to the mean curvature flow. Ann. Probab. 39(4), 1305–1331 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    K. Hara, Y. Takahashi, Lagrangian for pinned diffusion process, in Itô’s Stochastic Calculus and Probability Theory (Springer, Tokyo, 1996), pp. 117–128Google Scholar
  5. 5.
    N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, vol. 24, 2nd edn. (North-Holland, Amsterdam, 1989)Google Scholar
  6. 6.
    J. Lott, Optimal transport and Perelman’s reduced volume. arXiv:0804.0343v2Google Scholar
  7. 7.
    Y. Takahashi, S. Watanabe, The probability functionals (Onsager-Machlup functions) of diffusion processes, in Stochastic Integrals, (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Mathematics, vol. 851 (Springer, Berlin, 1981), pp. 433–463Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut Élie Cartan de LorraineVandœuvre-lès-NancyFrance

Personalised recommendations