Wave equation with stochastic boundary values
Consider the transverse displacement \(U(x,t)\) satisfying the linear partial differential equation \[ U_{tt}(x,t)=U_{xx}(x,t) \quad \text{on } 0 \leq x \leq L,\;0 \leq t<\infty, \tag{*} \] with stochastic boundary values of the form \[ \begin{cases} U(0,t)=0 \quad & \text{ on } 0 \leq t<\infty, \\ U_ x(L,t)=\dot B(t) & \text{ on } 0 \leq t<\infty, \\ U_ t(L,t)=\dot B(t) & \text{ on } 0 \leq t<\infty, \end{cases} \tag{**} \] where \(\dot B(t)\) is white noise. The authors prove that the solution of equation (*) with initial data of the form (**) satisfies the asymptotic estimates \[ \pi /2 \sqrt 2 \leq \liminf_{n \to \infty} \bigl( \| U(\cdot,t_ n)\|/ \sqrt{t_ n/ \log t_ n} \bigr) \quad\text{ (a.s.)} \] and \[ \limsup_{n \to \infty} \bigl( \| U(\cdot,t_ n)\|/ \sqrt {t_ n \log t_ n} \bigr) \leq \sqrt 2 \quad\text{ (a.s.)}, \] where \(\| U(\cdot,t)\|=\max_{0 \leq x \leq L} | U(x,t)|\) and \(\{t_ n\}\) is any suitable sequence of real numbers.
scientific article; zbMATH DE number 480445