On stochastic evolution equations with stochastic boundary conditions. (English. Russian original) Zbl 0802.60055
Theory Probab. Appl. 38, No. 1, 1-13 (1993); translation from Teor. Veroyatn. Primen. 38, No. 1, 3-19 (1993).
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.
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.
Reviewer: T.Bojdecki (Guanajuato)
MSC:
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
60G20 | Generalized stochastic processes |