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


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60G17 Sample path properties
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