Springer eBooks may be purchased by end-customers only and are sold without copy protection (DRM free). Instead, all eBooks include personalized watermarks. This means you can read the Springer eBooks across numerous devices such as Laptops, eReaders, and tablets.
You can pay for Springer eBooks with Visa, Mastercard, American Express or Paypal.
After the purchase you can directly download the eBook file or read it online in our Springer eBook Reader. Furthermore your eBook will be stored in your MySpringer account. So you can always re-download your eBooks.
This volume is devoted to a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel' boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called a Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel' boundary condition, on the boundary of the domain. Probabilistically, a Markovian particle moves both by jumps and continuously in the state space and it obeys the Ventcel' boundary condition, which consists of six terms corresponding to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the sticking (or viscosity) phenomenon, the jump phenomenon on the boundary, and the inward jump phenomenon from the boundary. In particular, second-order elliptic differential operators are called diffusion operators and describe analytically strong Markov processes with continuous paths in the state space such as Brownian motion. We observe that second-order elliptic differential operators with smooth coefficients arise naturally in connection with the problem of construction of Markov processes in probability. Since second-order elliptic differential operators are pseudo-differential operators, we can make use of the theory of pseudo-differential operators as in the previous book: Semigroups, boundary value problems and Markov processes (Springer-Verlag, 2004).
Our approach here is distinguished by its extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. Several recent developments in the theory of singular integrals have made further progress in the study of elliptic boundary value problems and hence in the study of Markov processes possible. The presentation of these new results is the main purpose of this book.
Content Level »Research
Keywords »Boundary value problem - Feller semigroup - Markov process - Probability theory - analytic semigroup - elliptic boundary value problem - semilinear parabolic equation