Randomly perturbed vibrations

Author:

Elshamy M.

Abstract

Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

Publisher

Cambridge University Press (CUP)

Subject

Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability

Reference10 articles.

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3. Random Perturbations of Dynamical Systems

4. Large Deviations for a Reaction-Diffusion Equation with Non-Gaussian Perturbations

5. Random perturbations of reaction-diffusion equations: the quasi-deterministic approach;Freidlin;Trans. Amer. Math. Soc.,1988

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