Averages in curvilinear boundaries of stochastic hyperbolic systems (Q919711)

Let \(\eta_{\varepsilon}(x,y)=\varepsilon^{-1/2}(\xi_{\varepsilon}(x\varepsilon^{-1/2},y\varepsilon^{-1/2})-u(x,y))\), where the random field \(\xi_{\varepsilon}(x,y)\) is a solution of the equation \[ \xi_{\varepsilon}(x,y)=\alpha (x)+\beta (y)+\varepsilon \int^x_0 \int^y_0 a(s,t,\xi_{\varepsilon}(s,t))\,ds\,dt+\varepsilon w(x,y), \] \[ \xi_{\varepsilon}(x,0)=\alpha(x),\quad \xi_{\varepsilon}(0,y)=\beta(y),\quad \alpha(0)=\beta(0), \] and \(u(x,y)\) is a solution of the determinate equation \[ \partial^ 2u(x,y)/\partial x\partial y=a_0(u(x,y)),\quad u(x,0)=\alpha(x),\quad u(0,y)=\beta(y). \] Here \(w(x,y)\) is a Wiener field on the plane and \(\varepsilon >0\) is a small parameter. Under some assumptions the authors estimate the rate of convergence of \(\eta_{\varepsilon}(x,y)\) to a solution of the linear stochastic equation of hyperbolic type \[ \eta (x,y)=\int^x_0 \int^y_0 G(u(s,t))\eta(s,t)\,ds\,dt+w(x,y). \] The proof is only outlined.