Abstract
We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.
The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.
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Röckle, H. Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients. Acta Applicandae Mathematicae 47, 323–349 (1997). https://doi.org/10.1023/A:1017906903335
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DOI: https://doi.org/10.1023/A:1017906903335