Advertisement

The vibrating string forced by white noise

  • E. M. Cabaña
Article

Summary

The equation of the vibrating string forced by white noise is formally solved, using stochastic integrals with respect to a plane Brownian motion, and it is proved that a certain process associated to the energy is a martingale. Then Doob's martingale inequality is used to furnish some probability bounds for the energy.

Such bounds provide a solution for the double barrier problem for the class of Gaussian stationary processes which can be represented as linear functionals of the positions and the velocities of the string.

Keywords

Stochastic Process Brownian Motion Stationary Process White Noise Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cabaña, E. M.: On stochastic differentials in Hilbert spaces. Proc. Amer. math. Soc. 20, 259–265 (1969).Google Scholar
  2. 2.
    Ciesielski, Z.: Hölder condition for realizations of Gaussian processes. Trans. Amer. math. Soc. 99, 403–413 (1961).Google Scholar
  3. 3.
    Feller, W.: Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1, 459–504 (1957).Google Scholar
  4. 4.
    —: On the equation of the vibrating string. J. Math. Mech. 8, 339–348 (1959).Google Scholar
  5. 5.
    ItÔ, K., McKean, H. P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965.Google Scholar
  6. 6.
    McKean, H. P.: Elementary solutions for certain parabolic partial differential equations. Trans. Amer. math. Soc. 82, 519–548 (1956).Google Scholar
  7. 7.
    — Stochastic integrals. New York: Academic Press (to appear).Google Scholar

Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • E. M. Cabaña
    • 1
  1. 1.Instituto de Matemática y EstadísticaUniversidad de la RepÚblicaMontevideoUruguay

Personalised recommendations