On existence and uniqueness of the solution of a boundary value problem for a parabolic equation with random perturbations (Q2784983)

Let \(\eta(t)\), \(t\geq 0,\) be a stationary, in the restricted sense zero mean random process such that \(E|\eta(s)|^2=b^2\). Let \(D\) be a bounded open set in \(R^{n}\), let \(\partial D\in C^2\) be the boundary of \(D\). The author proves an existence and uniqueness theorem for a solution from the space \(\overline{W}^{1,2}_2(D_{T})\) of the boundary value problem NEWLINE\[NEWLINE{\partial U_{\varepsilon}(t,x)\over\partial t}=L_{t,x} U_{\varepsilon}(t,x)+A(t,x,U_{\varepsilon}(t,x))+B(t,x,U_{\varepsilon}(t,x)) \eta(t/\varepsilon),NEWLINE\]NEWLINE NEWLINE\[NEWLINEU_{\varepsilon}(0,x)=u_0(x),\quad U_{\varepsilon}(t,x)|_{x\in\partial D}=0, NEWLINE\]NEWLINE where NEWLINE\[NEWLINEL_{t,x}U=\sum_{i,j=1}^{n}{\partial\over\partial x_{i}}\left[a_{i,j}(t,x){\partial U\over\partial x_{j}}\right]+\sum_{i=1}^{n}b_{i}(t,x){\partial U \over\partial x_{i}}NEWLINE\]NEWLINE and \(\varepsilon>0\) is a small parameter.