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Introduction and Main Results

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Semigroups, Boundary Value Problems and Markov Processes

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Abstract

In this monograph we solve the problem of the existence of Feller semigroups associated with strong Markov processes. More precisely, we prove the unique solvability of boundary value problems for Waldenfels integro-differential operators with general Ventcel’ (Wentzell) boundary conditions, and construct Feller semigroups corresponding to the diffusion phenomenon where a Markovian particle moves chaotically in the state space, incessantly changing its direction of motion until it “dies” at the time when it reaches the set where the particle is definitely absorbed. This monograph provides a careful and accessible exposition of the functional analytic approach to the problem of constructing strong Markov processes with Ventcel’ boundary conditions in probability. Our approach here is distinguished by the extensive use of ideas and techniques characteristic of recent developments in the theory of partial differential equations.

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  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Kazuaki Taira

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  1. Kazuaki Taira
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Taira, K. (2014). Introduction and Main Results. 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_1

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