Book contents
- Frontmatter
- Contents
- Preface
- Basic definitions, notation and abbreviations
- 1 Introduction
- Part I Tools from Markov process theory
- Part II Nonlinear Markov processes and semigroups
- Part III Applications to interacting particles
- Appendices
- A Distances on measures
- B Topology on càdlàg paths
- C Convergence of processes in Skorohod spaces
- D Vector-valued ODEs
- E Pseudo-differential operator notation
- F Variational derivatives
- G Geometry of collisions
- H A combinatorial lemma
- I Approximation of infinite-dimensional functions
- J Bogolyubov chains, generating functionals and Fock-space calculus
- K Infinite-dimensional Riccati equations
- References
K - Infinite-dimensional Riccati equations
from Appendices
Published online by Cambridge University Press: 05 July 2014
- Frontmatter
- Contents
- Preface
- Basic definitions, notation and abbreviations
- 1 Introduction
- Part I Tools from Markov process theory
- Part II Nonlinear Markov processes and semigroups
- Part III Applications to interacting particles
- Appendices
- A Distances on measures
- B Topology on càdlàg paths
- C Convergence of processes in Skorohod spaces
- D Vector-valued ODEs
- E Pseudo-differential operator notation
- F Variational derivatives
- G Geometry of collisions
- H A combinatorial lemma
- I Approximation of infinite-dimensional functions
- J Bogolyubov chains, generating functionals and Fock-space calculus
- K Infinite-dimensional Riccati equations
- References
Summary
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.
- Type
- Chapter
- Information
- Nonlinear Markov Processes and Kinetic Equations , pp. 355 - 359Publisher: Cambridge University PressPrint publication year: 2010