Abstract stochastic equations. II: Solutions in spaces of abstract stochastic distributions.
[For part I see \textit{I. V. Melnikova}, \textit{A. I. Filinkov} and \textit{U. A. Anufrieva}, ibid. 111, No. 2, 3430--3475 (2002; Zbl 1007.60059).] Consider the stochastic heat equation \[ {dX(t,x)\over dt}= \Delta_x X(t,x)+ {\mathcal W}(t,x),\quad t\in [0,1],\tag{1} \] where \(x\in {\mathcal O}= \{x\in\mathbb{R}^N;\,0< x_k< a_k,\,k= 1,\dots, N\}\), \(X(t,x)= 0\), \(t\in [0,T]\), \(x\in\partial{\mathcal O}\), \(X(0,x)= 0\), \(x\in{\mathcal O}\), with white noise \(\{{\mathcal W}(t,x),\,t\geq 0\}\). The approach of this paper allows us to obtain a solution of (1) in the case \(N> 1\) and the space \(S(H)_{-0}\) of stochastic distributions with values in \(H\). The central point of this paper is the construction of the spaces of \(H\)-valued test functions \(S(H)_\rho\), \(\rho\in [0,1]\), and stochastic distributions \(S(H)_{-\rho}\): \[ S(H)_1\subset S(H)_\rho\subset S(H)_0\subset L_2(S'; H)\subset S(H)_{-0}\subset S(H)_{-\rho}\subset S(H)_{-1} \] for a seperable Hilbert space \(H\). Then the authors develop the calculus in these spaces and consider basic examples of \(H\)-valued stochastic processes; the \(H\)-valued weak Wiener process \(\{W(t)\}\) and the \(H\)-valued singular white noise process \(\{{\mathcal W}(t)\}\). Generally, both of then take values in \(S(H)_{-0}\) for any fixed \(t\). One can expect that the \(Q\)-Wiener processes discussed in Part I belong to \(L_2(S';H)\). It is crucial for our main results on evolution equations that the white noise be (infinitely) differentiable with respect to \(t\) (\(t\) plays the role of a parameter) in \(S(H)_{-1}\), and in this space, we can give a description of the convergence similar to the convergence in \(S(\mathbb{R}^N)_{-1}\).
scientific article; zbMATH DE number 2094863