Abstract
In this chapter we introduce a class of (temporally homogeneous) Markov processes which we will deal with in this book (Definition 9.3). Intuitively, the Markov property is that the prediction of subsequent motion of a physical particle, knowing its position at time t, depends neither on the value of t nor on what has been observed during the time interval [0, t); that is, a physical particle “starts afresh”. In Sect. 9.2 we introduce a class of semigroups associated with Markov processes, called Feller semigroups, and give a characterization of Feller semigroups in terms of Markov transition functions. In Sects. 9.4 and 9.5, we reduce the problem of existence of Feller semigroups to the unique solvability of the boundary value problem for Waldenfels integro-differential operators W with Ventcel’ boundary conditions L in the theory of partial differential equations.
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Taira, K. (2014). Markov Processes, Transition Functions and Feller Semigroups. In: Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43696-7_9
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