Kolmogorov Operators in Spaces of Continuous Functions and Equations for Measures

This book presents a novel perspective of Markov semigroups, resulting in a better understanding of the relationships between Stochastic PDEs and Kolmogorov operators.

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.

Contenu

1. Introduction.- 2. Preliminaries.- 3. Measure valued equations for stochastically continuous Markov semigroups.- 4. Measure equations for Ornstein-Uhlenbeck operators.- 5. Bounded perturbations of Ornstein-Uhlenbeck operators.- 6. Lipschitz perturbations of Ornstein-Uhlenbeck operators.- 7. The reaction-diffusion operator.- 8. The Burgers equation.- Bibliography.- Index.