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.
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Cabaña, E.M. The vibrating string forced by white noise. Z. Wahrscheinlichkeitstheorie verw Gebiete 15, 111–130 (1970). https://doi.org/10.1007/BF00531880
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DOI: https://doi.org/10.1007/BF00531880