Analytical Methods for Kolmogorov Equations

Description

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. Dedication Contents Preface to the second edition Preface to the first edition About the author Introduction I Autonomous Kolmogorov equations 1 The elliptic equation and the Cauchy problem in Cb(RN): the uniformly elliptic case 2 One-dimensional theory 3 Uniqueness results, conservation of probability and maximum principles 4 Properties of T(t) in spaces of continuous functions 5 Uniform estimates for the derivatives of the function T(t)f 6 Pointwise estimates for the derivatives of the function T(t)f 7 Markov semigroups in Lp-spaces 8 Estimates on the Green function 9 The invariant measure μ and the semigroup in Lp(RN,μ) 10 The Ornstein-Uhlenbeck operator 11 Degenerate Markov semigroups in RN 12 The Cauchy-Dirichlet problem 13 The Cauchy-Neumann problem II Non-autonomous Kolmogorov equations 14 The evolution operator and the evolution semigroup in the space of bounded and continuous functions 15 Estimates for Green functions 16 The evolution operator in Lp-spaces 17 The evolution semigroups {T (t)} and {T ♯(t)} in Lp-spaces 18 The asymptotic behaviour of the evolution operator and the evolution semigroup III Appendices A Function spaces and smooth domains B Basic notions of functional analysis in Banach spaces C An overview of strongly continuous and analytic semigroups D PDE’s and analytic semigroups Bibliography Index