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from Appendices
Published online by Cambridge University Press: 05 July 2014
The construction of the Ornstein-Uhlenbeck (OU) semigroups from Section 10.4 is very straightforward. However, the corresponding process is Gaussian; hence it is also quite natural and insightful to construct infinite-dimensional OU semigroups and/or propagators alternatively, via the completion from its action on Gaussian test functions. In analyzing the latter, the Riccati equation appears. We shall sketch here this approach to the analysis of infinite-dimensional OU semigroups, starting with the theory of differential Riccati equations on symmetric operators in Banach spaces.
Let B and B* be a real Banach space and its dual, duality being denoted as usual by (., .). Let us say that a densely defined operator C from B to B* (that is possibly unbounded) is symmetric (resp. positive) if (Cν, ω) = (Cω, ν) (resp. if (Cν, ν) ≥ 0) for all ν, ω from the domain of C. By SL+(B, B*) let us denote the space of bounded positive operators taking B to B*. Analogous definitions are applied to the operators taking B* to B. The notion of positivity induces a (partial) order relation on the space of symmetric operators.
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