Asymptotic property of solutions to the random Cauchy problem for wave equations
The author considers random wave equations for the form \((*)\quad (d^ 2u/dt^ 2)(t,\omega)+S(\omega)^ 2u(t,\omega)=0\) (or \(=f(t,\omega))\) with initial conditions \((**)\quad u(0,\omega)=\phi_ 0(\omega),\) \((du/dt)(0,\omega)=\phi_ 1(\omega).\) Here \(\omega\in \Omega\), a probability space, with expectation denoted by E, and S(\(\omega)\) is a (measurable in \(\omega)\) self-adjoint operator on a Hilbert space H for each \(\omega\in \Omega\). Define the kinetic, potential, and total energies respectively by \(K(t,\omega)=\| du/dt\|^ 2_ H,\) \(P(t,\omega)=\| S(\omega)u\|^ 2_ H,\) \(Y_{\phi}(\omega)=K(t,\omega)+P(t,\omega)\) (independent of t). A typical result is \[ \lim_{t\to \pm \infty}E(K(t))=\lim_{t\to \pm \infty}E(P(t))=E(Y_{\phi}) \] for all solutions of (*), (**) and all \(\phi =(\phi_ 0,\phi_ 1)\) if and only if \[ \lim_{t\to \pm \infty}E\int^{\infty}_{0}e^{it\lambda}d\| F_{\lambda}(\omega)g\|^ 2=0 \] holds for all \(g\in H\) where \(S(\omega)=\int \lambda dF_{\lambda}(\omega)\) is the (integral) spectral representation of S(\(\omega)\). This is a stochastic version of a result of the reviewer [J. Math. Anal. Appl. 32, 392-399 (1970; Zbl 0216.418)], and the proof is based on that of the reviewer. Several related results and remarks are given.
scientific article; zbMATH DE number 3958376
scientific article