Lorenzi L. Analytical Methods for Kolmogorov Equations

New York: Chapman and Hall/CRC, 2016. — 607 p.The second edition of this book has a new title that more accurately reflects the table of contents. Over the past few years, many new results have been proven in the field of partial differential equations. This edition takes those new results into account, in particular the study of nonautonomous operators with unbounded coefficients, which has received great attention. Additionally, this edition is the first to use a unified approach to contain the new results in a singular place.Preface to the second edition
Preface to the first edition
About the authorAutonomous Kolmogorov equations
The elliptic equation and the Cauchy problem in Cb(RN): the uniformly elliptic case
One-dimensional theory
Uniqueness results, conservation of probability and maximum principles
Properties of T(t) in spaces of continuous functions
Uniform estimates for the derivatives of the function T(t)f
Pointwise estimates for the derivatives of the function T(t)f
Markov semigroups in Lp-spaces
Estimates on the Green function
The invariant measure μ and the semigroup in Lp(RN,μ)
The Ornstein-Uhlenbeck operator
Degenerate Markov semigroups in RN
The Cauchy-Dirichlet problem
The Cauchy-Neumann problem
Non-autonomous Kolmogorov equations
The evolution operator and the evolution semigroup in the space of bounded and continuous functions
Estimates for Green functions
The evolution operator in Lp-spaces
The evolution semigroups {T (t)} and {T ♯(t)} in Lp-spaces
The asymptotic behaviour of the evolution operator and the evolution semigroup
Appendices
Function spaces and smooth domains
Basic notions of functional analysis in Banach spaces
An overview of strongly continuous and analytic semigroups
PDE’s and analytic semigroups