For the first time in book form, "Analytical Methods for Markov Semigroups" provides a comprehensive analysis on Markov semigroups both in spaces of bounded and continuous functions as well as in Lp spaces relevant to the invariant measure of the semigroup. Exploring specific techniques and results, the book collects and updates the literature associated with Markov semigroups. Divided into four parts, the book begins with the general properties of the semigroup in spaces of continuous functions: the existence of solutions to the elliptic and to the parabolic equation, uniqueness properties and counterexamples to uniqueness, and the definition and properties of the weak generator. It also examines properties of the Markov process and the connection with the uniqueness of the solutions. In the second part, the authors consider the replacement of RN with an open and unbounded domain of RN. They also discuss homogeneous Dirichlet and Neumann boundary conditions associated with the operator A. The final chapters analyze degenerate elliptic operators A and offer solutions to the problem.Using analytical methods, this book presents past and present results of Markov semigroups, making it
Introduction
MARKOV SEMIGROUPS IN RN
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 T(t)f
Pointwise Estimates for the Derivatives of T(t)f
Invariant Measures µ and the Semigroup in LP(RN, µ)
The Ornstein-Uhlenbeck Operator
A Class of Nonanalytic Markov Semigroups in Cb(RN) and in Lp(RN, µ)
MARKOV SEMIGROUPS IN UNBOUNDED OPEN SETS
The Cauchy-Dirichlet Problem
The Cauchy-Neumann Problem: The Convex Case
The Cauchy-Neumann Problem: The Nonconvex Case
A CLASS OF MARKOV SEMIGROUPS IN RN ASSOCIATED WITH DEGENERATE ELLIPTIC OPERATORS
The Cauchy Problem in Cb(RN)
APPENDICES
Basic Notions of Functional Analysis in Banach Spaces
An Overview on Strongly Continuous and Analytic Semigroups
PDE's and Analytic Semigroups
Some Properties of the Distance Function
Function Spaces: Definitions and Main Properties
References
Index
