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Second-order neutral stochastic evolution equations with heredity. (English) Zbl 1077.34062

The author considers an abstract neutral semilinear stochastic evolution equation of the form \[ d(X'(t)-f_1(t,X_t)) =AX(t)dt+f_2(t,X_t)dt+g(t,X_t)dW(t). \] Here, \(A\) generates a strongly continuous cosine family of operators on a Hilbert space, \(W\) is an infinite-dimensional Wiener process and \(X_t\) denotes the solution segment at time \(t\). Various results for this equation are derived. First, existence and uniqueness of a mild solution is established under Lipschitz conditions. Then, continuous dependenc on the initial segment as well as moment estimates are treated. Finally, the existence and uniqueness result is generalized to the case of Carathéodory conditions. The paper closes with an example discussing an initial-boundary value problem.

MSC:

34F05 Ordinary differential equations and systems with randomness
34K50 Stochastic functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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