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Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. (English) Zbl 0766.60078
Let \({\mathcal O}\subset\mathbb{R}^ d\) be some domain and \(\mu\) be a finite measure on the Borel \(\sigma\)-algebra of \({\mathcal O}\). The author studies the stochastic integral equation \[ X(t)=U(t,0)X_ 0+\int^ t_ 0U(t,s)R(s,X(s))ds+\int^ t_ 0U(t,s)F(s,X(s))dW_ s \] for \(t>0\), where \(U\) is a two-parameter strongly continuous semigroup, and \(W\) denotes an \(\mathbb{L}_ 2({\mathcal O},d\mu)\)-valued Brownian motion (regular as well as cylindrical). The main purpose of this paper is to establish existence, uniqueness (Lipschitz case, linear growth) and smoothness of a solution to this equation. To obtain the last a multiparameter approach is used and the relation of this approach to the one used before in this paper is discussed. Many of the results are known in special cases, but not in the general situation considered by the author.
Reviewer: R.Manthey (Jena)

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