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On a stochastic integrodifferential evolution equation of Volterra type. (English) Zbl 0583.60060

We initiate a study of the stochastic integrodifferential evolution equation \[ \dot x(t,\omega)=Ax(t,\omega)+\int^{t}_{0}B(t- s)x(s,\omega)d\beta (s,\omega)+f(t) \] of Volterra type in a Hilbert space set up, where \(\beta\) (t) is a Hilbert-Schmidt operator valued Brownian motion. We first present some existence-uniqueness theorems. Next, we consider approximation theorems of the following types with mean-square convergence uniformly on compacta:
(1) Approximate unbounded operators \(A\) and \(B\) by Yosida type approximations with bounded operators; (2) Approximate the generator A by generators \(A_ n\), and \(B\) by a sequence \(B_ n\); and (3) Zeroth order approximation of the stochastic equation by a deterministic evolution equation \(\dot y(\)t)\(=Ay(t)+f(t).\)
We also prove the weak convergence of measures induced by solutions of approximate equations to the measure induced by the solution of the given equation.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
60B10 Convergence of probability measures
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