Second Order PDE’s in Finite and Infinite Dimension

A Probabilistic Approach

  • Sandra Cerrai

Part of the Lecture Notes in Mathematics book series (LNM, volume 1762)

About this book

Introduction

The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results, we study the associated Kolmogorov equations, the large-time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton- Jacobi-Bellman equations. In the literature there exists a large number of works (mostly in finite dimen­ sion) dealing with these arguments in the case of bounded Lipschitz-continuous coefficients and some of them concern the case of coefficients having linear growth. Few papers concern the case of non-Lipschitz coefficients, but they are mainly re­ lated to the study of the existence and the uniqueness of solutions for the stochastic system. Actually, the study of any further properties of those systems, such as their regularizing properties or their ergodicity, seems not to be developed widely enough. With these notes we try to cover this gap.

Keywords

Kolmogorov equations Parameter diffusion process ergodicity partial differential equation

Editors and affiliations

  • Sandra Cerrai
    • 1
  1. 1.Dipartimento di Matematica per le DecisioniUniversità di FirenzeFirenzeItaly

Bibliographic information

  • DOI https://doi.org/10.1007/b80743
  • Copyright Information Springer-Verlag Berlin Heidelberg 2001
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-42136-8
  • Online ISBN 978-3-540-45147-1
  • Series Print ISSN 0075-8434
  • About this book