Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

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Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise (2003, background): “9) and the theory of stochastic integration in separable Banach spaces as developed in [2].”
γ-Radonifying Operators: A Survey (2010, unknown): “Building on previous work of Da Prato andZabczyk [27] and van Neerven and Brzeźniak [14], it has been shown in van Neerven and Weis [90] that the linear stochastic Cauchy problem dU(t) = AU(t) dt+B dWH(t), t > 0, U(0) = u0, admits a unique weak solution U if and only if for some (and then for all) T”
Stochastic integration of functions with values in a Banach space (2005, methodology): “These were extended to separable Banach spaces E in [4] with an indirect method of proof; see also the remarks in the introduction.”
Maximal regularity for stochastic convolutions driven by Lévy processes (2009, background): “[ 8 ] for a discussion of this subject, as a solution to the following E–valued Langevin equation written in the Ito-form”
Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes (2003, unknown): “Assume that S(t)H ⊆ H for all t ≥ 0.”
Stochastic integrals for spde's: a comparison (2010, background): “We do not discuss Banach-space-valued stochastic integrals, for which we refer to [2] and [40].”