A method from stochastic flow theory is used to obtain smoothing properties of the transition semigroup Pt of a class of stochastic differential equations on Hilbert space. The equations considered may have unbounded coefficients and include such stochastic partial differential equations as for Xt in L2 (0,π) In certain cases a formula for the Fréchet derivative of Ptf is given, exhibiting this smoothing property
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Prato et al. (1995) studied this question.