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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].

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
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