ABSTRACT
Wave motion and mechanical vibration are two of the most commonly observed physical phenomena. As mathematical models, they are usually described by partial differential equations of hyperbolic type. The most wellknown one is the wave equation. In contrast with the heat equation, the firstorder time derivative term is replaced by a second-order one. Excited by a white noise, a stochastic wave equation takes the form
∂2u
∂t2 = c2∆u+ σ0W˙ (x, t)
in some domain D, where c is known as the wave speed and σ0 is the noise intensity parameter. If the domain D is bounded in Rd, the equation can describe the random vibration of an elastic string for d = 1, an elastic membrane for d = 2, and a rubbery solid for d = 3. When D = Rd, it may be used to model the propagation of acoustic or optical waves generated by a random excitation. For a large-amplitude vibration or wave motion, some nonlinear effects must be taken into consideration. In this case we should include appropriate nonlinear terms in the wave equation. For instance, it may look like
∂2u
∂t2 = c2∆u+ f(u) + σ(u)W˙ (x, t),
where f(u) and σ(u) are some nonlinear functions of u. More generally, the wave motion may be modeled by a linear or nonlinear first-order hyperbolic system. A special class of stochastic hyperbolic system will be discussed later on.
