Overview
- Presents a novel perspective of Markov semigroups, resulting in a better understanding of the relationships between Stochastic PDEs and Kolmogorov operators
- Special attention is paid to well-known models as the Ornstein-Uhlenbeck semigroup, the reaction-diffusion and Burgers equations SPDE. For each of them, the associated Kolmogorov operator is considered and the Kolmogorov equation for measures is solved. Moreover, a new characterization of the generator is given in the space if continuous functions by means of a core.
- Most of the results which can be found in the literature are obtained in functional spaces weighted by a suitable invariant measure for the transition semigroup. In this book no assumptions about ergodicity are given, and many properties of the transition semigroup and of the Kolmogorov operator are studied in spaces of continuous functions.
- The difficult problem of showing the uniqueness of a solution of the Kolmogorov equation for measure (also called Fokker-Planck or Kolmogorov backward) is solved.
Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 10)
Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)
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Keywords
- Kolmogorov operator
- Markov semigroup
- Ornstein-Uhlenbeck operator
- Stochastic partial differential equation
- partial differential equations
About this book
The book is devoted to study the relationships between Stochastic Partial Differential Equations and the associated Kolmogorov operator in spaces of continuous functions.
In the first part, the theory of a weak convergence of functions is developed in order to give general results about Markov semigroups and their generator.
In the second part, concrete models of Markov semigroups deriving from Stochastic PDEs are studied. In particular, Ornstein-Uhlenbeck, reaction-diffusion and Burgers equations have been considered. For each case the transition semigroup and its infinitesimal generator have been investigated in a suitable space of continuous functions.
The main results show that the set of exponential functions provides a core for the Kolmogorov operator. As a consequence, the uniqueness of the Kolmogorov equation for measures has been proved.
Reviews
From the reviews:
“This volume of the Edizioni Della Normale is based on the authors’s thesis about the relationship of the solutions of a class stochastic partial differential equations (SPDE) with additive noise, the associated Kolmogorov operator and the associated Markov generator on spaces of continuous functions. … This monography gives a nice introduction to the field of measure valued Kolmogorov equations for a quite general class SPDE.” (Michael Högele, Zentralblatt MATH, Vol. 1198, 2010)Authors and Affiliations
Bibliographic Information
Book Title: Kolmogorov Operators in Spaces of Continuous Functions and Equations for Measures
Authors: Luigi Manca
Series Title: Publications of the Scuola Normale Superiore
Publisher: Edizioni della Normale Pisa
Copyright Information: Edizioni della Normale 2009
Softcover ISBN: 978-88-7642-336-9Published: 29 December 2008
Series ISSN: 2239-1460
Series E-ISSN: 2532-1668
Edition Number: 1
Number of Pages: 130