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Remarks on the existence and approximation for semilinear stochastic differential equations in Hilbert spaces. (English) Zbl 1002.60058

Let \(H\) be a real separable Hilbert space, \(A\) an infinitesimal generator of a \(C_0\)-semigroup on \(H\), \(W\) a cylindrical \(Q\)-Wiener process on \(H\). A stochastic evolution equation \[ dX(t) = (AX(t) + f(X(t))) dt + g(X(t)) dW(t),\quad X(0)=x, \] is studied. First, it is shown that existence and uniqueness of a mild solution to (1) may be established by the method of successive approximations under assumptions upon \(f\) and \(g\) which are weaker than Lipschitz continuity and were proposed, in the finite-dimensional case, by T. Taniguchi [J. Differ. Equations 96, No. 1, 152-169 (1992; Zbl 0744.34052)]. In the second part of the paper, it is supposed that the semigroup generated by \(A\) is compact, but the functions \(f\) and \(g\) are merely continuous, of a linear growth, and such that pathwise uniqueness holds for (1). It is proven that Euler and Lie-Trotter approximations converge to the solution of (1) in the \(L^{p}\)-norm and that the solution depends continuously on data and on the coefficients. Finally, equations with nonlinear terms \(f\), \(g\) defined only on an open subset of \(H\) are considered and existence of a solution to (1) is shown under a hypothesis that a suitable sequence of Lyapunov functions exists.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 0744.34052
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