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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

$dX\left(t\right)=\left(AX\left(t\right)+f\left(X\left(t\right)\right)\right)dt+g\left(X\left(t\right)\right)dW\left(t\right),\phantom{\rule{1.em}{0ex}}X\left(0\right)=x,$

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