Let \(I= (t_ 0, t_ 1)\) and \(G\) be a region in \(\mathbb{R}^ d\). The paper deals with a linear stochastic equation \[ d\xi_ t= A\xi_ t dt+ d\eta_ t, \qquad t\in I, \tag{*} \] where \(A\) is a symmetric elliptic differential operator of the form \(A= \sum_{| k|\leq 2p} a_ k \partial^ k\), \(d\eta\) is of the white noise type and the solution \(\xi\) is looked for a class of distribution-valued processes. More precisely, one defines a space of generalized test functions \(X(I\times G)\) as the closure of \(C_ 0^ \infty (I\times G)\) in the norm of the conjugate Sobolev space \(W_ 2^{-(1,2p)}\), and \(\xi= (\xi_ t)_{t\in I}\), acting on functions from \(C_ 0^ \infty (I\times G)\) by means of the formula \((\varphi,\xi)= \int_ I (\varphi_ t, \xi_ t)dt\), should be mean-square continuous with respect to this norm. The space of random elements with this property is denoted by \({\mathbf W}(I\times G)\). If \(\xi\in {\mathbf W} (I\times G)\), then \((x,\xi)\) is well-defined for any \(x\in X(I\times G)\). Now, random boundary conditions on the solution of (*) can be imposed. They are of the form \[ \begin{alignedat}{2} (\delta\times x,\xi) &= (\delta\times x,\xi^ +), \qquad &&\delta\in C_ 0^ \infty(I), \quad x\in X_ A(\partial G),\\ (\delta_{t_ 0}\times x,\xi) &= (\delta_{t_ 0} \times x,\xi^ +), \qquad &&x\in C_ 0^ \infty(G), \end{alignedat} \] where \(\xi^ +\) is a fixed given element of \({\mathbf W}(I\times G)\), \(X_ A (\partial G)\) is an appropriate class of boundary test functions \((\text{supp } x\in \partial G)\), \(\delta_{t_ 0}\) is the usual delta function at \(t_ 0\). It is proved that under these boundary conditions, equation (*) has a unique solution (in \({\mathbf W} (I\times G)\)). If \(\eta\) and \(\xi^ +\) are independent, then this solution is Markov. Moreover, it is shown that if \(A\) is selfadjoint \(\leq 0\), then the formula \(\xi_ t= \int_{t_ 0}^ t e^{A(t-s)} d\eta_ s\) gives the (unique) solution of (*) for \(\xi^ + =0\).
It should be remarked that the English version of the paper is not a literal translation of the Russian one. The original (Russian) paper contains more explanations and therefore it is easier to read.