Da Prato, Giuseppe An introduction to Markov semigroups. (English) Zbl 1077.47044 Iannelli, Mimmo (ed.) et al., Functional analytic methods for evolution equations. Based on lectures given at the autumn school on evolution equations and semigroups, Levico Terme, Trento, Italy, October 28–November 2, 2001. Berlin: Springer (ISBN 3-540-23030-0/pbk). Lecture Notes in Mathematics 1855, 1-63 (2004). This is an excellent short lecture course for non-probabilists with knowledge of functional analysis and measure theory. Advantageously, it starts from Gaussian measures in a real separable Hilbert space \(H\), Gaussian random variables on \(H\), reproducing kernels and the Cameron-Martin formula used later. The main object then is a semigroup \((P_t)_{t\geq 0}\) of bounded positive linear integral operators acting on functions \(\varphi : H \to \mathbb R\), either continuous or Borel; \(P_t 1=1\), \(t\geq 0\). Also the mapping \([0,T]\times H \to \mathbb R\), \((t,x)\to P_t \varphi(x)\) is continuous (resp. Borel) for any \(\varphi\in C_b(H)\) (resp. \(\varphi\in B_b(H)\)) and \(T>0\). That framework permits to treat ergodicity, irreducibility, existence and uniqueness of invariant measures and the strong Feller property relevant to some important Markov semigroups. The examples considered include the heat semigroup in infinite dimensions, the Ornstein-Uhlenbeck semigroup and the diffusion semigroup arising from one-dimensional dynamics perturbed additively by (increments of) Brownian motion. For further information, the readers may consult the author’s previous works, especially his three joint monographs with J. Zabczyk.For the entire collection see [Zbl 1052.47002]. Reviewer: Andrej V. Bulinski (Moskva) Cited in 1 Document MSC: 47D07 Markov semigroups and applications to diffusion processes 47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory 60Jxx Markov processes 60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory Keywords:real separable Hilbert space; Gaussian measure; probability kernel; integral operator; Markov semigroup; ergodicity; invariant measures; Ornstein-Uhlenbeck semigroup; Brownian motion Citations:Zbl 0761.60052; Zbl 0849.60052; Zbl 1012.35001 PDFBibTeX XMLCite \textit{G. Da Prato}, Lect. Notes Math. 1855, 1--63 (2004; Zbl 1077.47044)