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A maximal inequality for stochastic convolution integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations. (English) Zbl 0622.60065
The author considers continuity properties of stochastic convolution integrals \[ Y(t)=\int^{t}_{0}U(t-s)dM(s), \] where U(\(\cdot)\) is an analytic semigroup on a complex separable Hilbert space H and M a cylindrical martingale. Let A be the generator of U(\(\cdot)\) and choose \(\beta >\) spectral radius of A. Assume that for some \(\gamma\in [0,)\) \((\beta -A)^{-\gamma}M\) is an H-valued martingale.
Under an integrability condition Y - considered as a stochastic process with values in the space \(dom((\beta -A)^{\alpha})\), endowed with the norm \(\| (\beta -A)^{\alpha}\cdot \|\)- has a.s. continuous paths if \(\alpha\) is sufficiently small (possibly 0). For suitable \(\alpha\) and \(\gamma\) Y is even Hölder continuous.
This generalizes the result of D. A. Dawson [Math. Biosci. 15, 287- 316 (1972; Zbl 0251.60040)], who proved continuity for the case where A is self-adjoint, \(A^{-1}\) Hilbert-Schmidt, M the cylindrical Wiener process, and \(\alpha =0\).
Reviewer: K.U.Schaumlöffel

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