A careful and accessible exposition of functional analytic methods in
stochastic analysis is provided in this book. It focuses on the
interrelationship between three subjects in analysis: Markov processes,
semi groups and elliptic boundary value problems. The author studies a
general class of elliptic boundary value problems for second-order,
Waldenfels integro-differential operators in partial differential
equations and proves that this class of elliptic boundary value problems
provides a general class of Feller semigroups in functional analysis. As
an application, the author constructs a general class of Markov
processes in probability in which a Markovian particle moves both by
jumps and continuously in the state space until it 'dies' at the time
when it reaches the set where the particle is definitely absorbed.
Augmenting the 1st edition published in 2004, this edition includes four
new chapters and eight re-worked and expanded chapters. It is amply
illustrated and all chapters are rounded off with Notes and Comments
where bibliographical references are primarily discussed. Thanks to the
kind feedback from many readers, some errors in the first edition have
been corrected. In order to keep the book up-to-date, new references
have been added to the bibliography. Researchers and graduate students
interested in PDEs, functional analysis and probability will find this
volume useful.