Approximation by time discretization of special stochastic evolution equations (Q2770659)

Let \(V\), \(H\) be separable Banach and Hilbert spaces, respectively, making a Gelfand triple \(V\subset H\subset V^*\), \(w(t)\) be a real-valued Wiener process. The author derives a time discretization scheme to construct a solution to the SPDE NEWLINE\[NEWLINE dX=-{A}X(t)dt+f(t,X(t))dt+g(t,X(t))dw,\quad X(0)=X_0\in H,\tag{1}NEWLINE\]NEWLINE under the assumption that \( A:V\to V^*\) is a maximal monotone operator while \(f\) and \(g\) are Lipschitz on \([0,T]\times H\) for fixed time interval \([0,T]\). To this end (1) is changed for an auxiliary equation NEWLINE\[NEWLINE d(e^{-\alpha t}X(t))=-e^{-\alpha t}[{ A}X(t)dt+ \alpha X(t)]dt+e^{-\alpha t}[f(t,X(t))dt+g(t,X(t))dw] NEWLINE\]NEWLINE where \(\alpha\) is a real positive number and the corresponding discretized problem NEWLINE\[NEWLINE \begin{gathered} e^{-\alpha t_n}(x_n+[{ A}x_n+\alpha x_n]h_n) = e^{-\alpha t_{n-1}}[y_{n-1} +f(t_{n-1}, x_{n-1})h_n],\\ y_n= x_n+g(t_{n-1}, x_n)[w(t_n)-w(t_{n-1})], \quad n=1,2, \dots , N,\quad x_0=y_0=X_0,\end{gathered} NEWLINE\]NEWLINE with \(x_n\in V\), \(y_n\in H\), is proved to have a unique solution \(x_n\in V\). Finally, the step process \(\hat x_N(t)=\sum_{n=1}^{N} I_{[t_{n-1}, t_n)}(t)x_n\) is proved to give rise to a sequence \(\hat x_N\) that strongly converges to \(X\) in the space \(L^2(\Omega\times [0,T], V)\) as \(N\to \infty\).