Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

Upper as well as lower bounds for both the large deviations and small deviations of several sup-norms associated with the displacements of a one-dimensional string driven by a Gaussian noise which is white in time and has general spatial covariance are developed.

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.