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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),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:
60H15Stochastic partial differential equations