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On Drifting Brownian Motion Made Periodic

  • Paul McGill
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

Abstract

The Brownian reference measure on periodic functions provides a framework for investigating more general circular processes. These include a significant class of periodic diffusions. We illustrate by proposing simple analytic criteria for finiteness and absolute continuity of the intrinsic circular measure associated to drifting Brownian motion. Our approach exploits a property of approximate bridges.

Keywords

Drifting Brownian motion Approximate bridge Circular measure Girsanov formula 

AMS Classification 2010

Primary: 60J65; Secondary: 60H10 28C20 

References

  1. 1.
    A. Borodin, P. Salminen, Handbook of Brownian Motion – Facts and Formulae (Birkhaüser, Basel, 2002)Google Scholar
  2. 2.
    S. Cambronero, Some Ornstein-Uhlenbeck potentials for the one-dimensional Schrödinger operator part II: position-dependent drift. Revista de Mathemática: Teoria y Applicationes 9(2), 31–38 (2002)Google Scholar
  3. 3.
    S. Cambronero, H.P. McKean, The ground state eigenvalue of Hill’s equation with white noise potential. Commun. Pure Appl. Math. 52(10), 1277–1294 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    K. Itô, H.P. McKean, Diffusion Processes and Their Sample Paths, 2nd edn. (Springer, Berlin, 1974)zbMATHGoogle Scholar
  5. 5.
    Th. Jeulin, Application de la théorie du grossissement à l’étude des temps locaux browniens, in Grossissement de filtrations: exemples et applications. Lecture Notes in Mathematics, vol. 1118 (Springer, Berlin, 1985), pp. 197–304Google Scholar
  6. 6.
    N.H. Kuiper, Tests concerning random points on a circle. Indag. Math. 22, 32–37, 38–47 (1960)CrossRefGoogle Scholar
  7. 7.
    H.P. McKean, K.T. Vaninsky, Statistical mechanics of nonlinear wave equations, in Trends and Perspectives in Applied Mathematics, ed. by L. Sirovich (Springer, Berlin, 1994), pp. 239–264CrossRefGoogle Scholar
  8. 8.
    A. Nahmod, L. Rey-Bellet, S. Sheffield, G. Staffilani, Absolute continuity of Brownian bridges under certain gauge transformations. Math. Res. Lett. 18(5), 875–887 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Quastel, B. Valko, KdV preserves white noise. Commun. Math. Phys. 277(3), 707–714 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefGoogle Scholar
  11. 11.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966)zbMATHGoogle Scholar
  12. 12.
    A.K. Zvonkin, A transformation of the phase space of a process that removes the drift. Math. USSR Sb. 22, 129–149 (1974)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Claude Bernard Lyon 1VilleurbanneFrance

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