On some problem of restoration of Wiener field on the plane (Q2703295)

Let \(w(x,y)\) be a Wiener field; \(\Pi=\{(x,y): a\leq x\leq b\), \(c\leq y\leq d\}\); \(\partial \Pi\) is the boundary of \(\Pi\). Observe the field \(w(x,y)\), \((x,y)\in\partial \Pi\). The best mean square estimate of \(w(x,y)\), \((x,y)\notin \partial\Pi\), is given by conditional expectation \(m=E[w(u,v)\mid {\mathcal F}_{\partial\Pi}]\). The mean square error of this estimate is \(d^{2}=E[[w(u,v)-m]^{2} \mid {\mathcal F}_{\partial\Pi}]\), where \({\mathcal F}_{\partial\Pi}= \sigma\{w(x,y): (x,y)\in\partial\Pi\}\). If \((u,v)\in\Pi\setminus\partial\Pi\), then with probability 1: NEWLINE\[NEWLINE\begin{multlined} m=-\alpha\beta w(a,c)+\beta w(u,c)-(1-\alpha)\beta w(b,c)+(1-\alpha)w(b,v)-(1-\alpha)(1-\beta)w(b,d)+\\ (1-\beta)w(u,d)- \alpha(1-\beta)w(a,d)+\alpha w(a,v),\end{multlined}NEWLINE\]NEWLINE NEWLINE\[NEWLINEd^{2}=\alpha\beta(u-a)(v-c),NEWLINE\]NEWLINE where \(\alpha=(b-u)/(b-a)\), \(\beta=(d-v)/(d-c)\). Analogous results for the regions \(S_{0}=\{(x,y):0\leq x<a,0\leq y<c\}\), \(S_{1}=\{(x,y):a\leq x\leq b,0\leq y\leq c\}\) and \(S_{2}=\{(x,y):0\leq x\leq a,c\leq y\leq d\}\) are obtained.