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On some regularity properties of solutions to stochastic evolution equations in Hilbert spaces. (English) Zbl 0704.60059
A linear stochastic equation $$dX=AX dt+dM$$ in a Hilbert space is considered, where M denotes a continuous, square integrable martingale and A is a generator of an analytic semigroup of bounded operators. By a solution to the above equation is meant the so-called mild solution given by the variation of constants formula.
Denote by $$H_{\alpha}$$ the space $$Dom((\lambda I-A)^{\alpha /2})$$, for an appropriately chosen $$\lambda$$, endowed with a graph norm. The main result of the paper describes the regularity of the solution X in the space $$H_{\alpha}$$. If M is a Wiener process then X is shown to be Hölder-continuous in $$H_{\alpha}$$ with an exponent smaller than (1- $$\alpha$$)/2 for any $$\alpha\in (0,1)$$. The proof follows from Kolmogorov’s test for continuity. The case of cylindrical martingales is also considered.
Reviewer: B.Gołdys
##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60G15 Gaussian processes 60G17 Sample path properties
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