Thismonographisanexpandedandrevisedversionofasetoflecturenotesfor thegraduatecoursesgivenbytheauthorbothatHiroshimaUniversity(1995- 1997) and at the University of Tsukuba (1998-2000)which were addressed to the advanced undergraduates and beginning-graduate students with interest in functional analysis, partial di?erential equations and probability. The ?rst edition of this monograph, which was based on the lecture notes given at the University of Tsukuba (1988-1990), was published in 1991. This edition was found useful by a number of people, but it went out of print after afewyears. Thissecondeditionhasbeenrevisedtostreamlinesomeoftheanalysisand to give better coverage of important examples and applications. The errors in the ?rst printing are corrected thanks to kind remarks of many friends. In order to make the monograph more up-to-date, additional references have been included in the bibliography. This secondedition may be consideredas a short introduction to the more advanced book "Semigroups, boundary value problems and Markov processes" which was published in the Springer Monographs in Mathematics series in 2004.For graduatestudents workingin functionalanalysis,partialdi?erential equations and probability, it may serve as an e? ective introduction to these three interrelated ?elds of analysis. For graduate students about to major in the subject and mathematicians in the ?eld looking for a coherent overview, it will provide a method for the analysis of elliptic boundary value problems p in the framework of L spaces. My special thanks go to the editorial sta?s of Springer-Verlag for their unfailing helpfulness and cooperation during the production of this second edition. This research was partially supported by Grant-in-Aid for General Sci- ti?cResearch(No.19540162),MinistryofEducation,Culture,Sports,Science and Technology, Japan.