part 1 Part I: Markov semigroups in RN -- chapter 2 The elliptic equation and the Cauchy problem in Cb(R N ): the uniformly elliptic case -- chapter 3 One-dimensional theory -- chapter 4 Uniqueness results, conservation of probability and maxi- mum principles -- chapter 5 Properties of {T (t)} in spaces of continuous functions -- chapter 6 Uniform estimates for the derivatives of T (t)f -- chapter 7 Pointwise estimates for the derivatives of T (t)f -- chapter 8 Invariant measures µ and the semigroup in Lp(RN, µ) -- chapter 9 The Ornstein-Uhlenbeck operator -- chapter 10 A class of nonanalytic Markov semigroups in Cb(R N ) and in Lp(RN, µ) -- part 2 Part II: Markov semigroups in unbounded open sets -- chapter 11 The Cauchy-Dirichlet problem -- chapter 12 The Cauchy-Neumann problem: the convex case -- chapter 13 The Cauchy-Neumann problem: the nonconvex case -- part 3 Part III: A class of Markov semigroups in RN associated with degenerate elliptic operators -- chapter 14 The Cauchy problem in Cb(R N ) -- part 4 Part IV: Appendices -- chapter A Basic notions of functional analysis in Banach spaces -- chapter B An overview on strongly continuous and analytic semigroups -- chapter 100 Appendix C: PDE’s and analytic semigroups -- chapter D Some properties of the distance function -- chapter E Function spaces: definitions and main properties E.1 Spaces of functions that are continuous or Ho¨lder continuous.