The vibrating string forced by white noise

Abstract

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.

Energy of the stochastic wave equation driven by a fractional Gaussian noise

“…It has been popular to consider the stochastic wave equation driven by a Gaussian noise. For example, the energy and displacement of a vibrating string forced by a space-time white noise is considered in [4] and [5]. The equation modeling this system is…”

Pathwise uniqueness for the stochastic heat equation with Hölder continuous drift and noise coefficients

“…We assume that W has the following covariance structure E(W t (φ)W t (ψ)) = t 0 R φ(s, y)ψ(s, y)dyds, t ≥ 0, for φ, ψ ∈ C ∞ c (R + × R). The stochastic heat equation with space-time white noise was studied among many others, by Cabaña [2], Dawson [8], [9], Krylov and Rozovskii [15], [17], [16], Funaki [14], [10] and Walsh [30]. Pathwise uniqueness of the solutions for the stochastic heat equation, when the white noise coefficient σ and the drift coefficient are Lipschitz continuous was derived in [30].…”

Pathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noise

“…(6.69) (6.69) is trivial in case (1). If y is between x and B(t − t ′ ), argue as in (6.64), this time using (t, x) ∈ Z(N ′ , n, K, β), to see that…”