Stabilization of the moment functions of the statistical solution of the wave equation (Q1091536)

One refers to the Cauchy problem for the wave equation \[ \partial^ 2u(t,x)/\partial t^ 2=a^ 2\Delta u(t,x),\quad x\in R^ 3,\quad t>0;\quad u(0,x)=u^ 0(x);\quad \partial u(0,x)/\partial t=u^ 1(x),\quad x\in R^ 3 \] where the initial conditions \(u=(u^ 0,u^ 1)\) belong to the space \[ {\mathcal H}_ 0=\{u| \quad u=(u^ 0,u^ 1);\quad D^{\alpha}u^ 0,D^{\beta}u^ 1\in L_{\infty}^{loc}(R^ 3);\quad | \alpha | \leq 2,\quad | \beta | \leq 1\} \] with finite seminorm. Let \(\mu_ 0(\omega)\) be a given measure on \({\mathcal H}_ 0\) defining a probability, such that \(u(x)=(u^ 0(x),u^ 1(x))\) belongs to the Borel-set \({\mathcal H}_ 0\). The main result of the paper is: under certain conditions regarding the measure \(\mu_ 0\) and for all \(n\in N\), \(\phi_ 1,\phi_ 2,...,\phi_ n\in C_ 0^{\infty}(R^ 3)\), \[ <M_ n^{{\bar \ell}^ n}(t,\bar x^ n),\phi_ 1(x_ 1)...\phi_ n(x_ n)>\to_{t\to \infty}<M_ n^{{\bar \ell}^ n}(\infty,\bar x^ n),\phi_ 1(x_ 1),...,\phi_ n(x_ n)>. \] With \(M_ n^{{\bar \ell}^ n}(\infty,x^ n)\) have been denoted the moments of the Gaussian measure \(\mu_{\infty}\) with zero expectation and a certain correlation function involving the fundamental solution of the Laplace-operator. There is no proof; one refers to the paper of \textit{N. E. Ratanov} [Russ. Math. Surv. 39, No.1, 179-180 (1984); translation from Usp. Mat. Nauk 39, No.1(235), 151-152 (1984; Zbl 0567.35051)].