04922nam a22005175i 4500 978-981-97-3659-1 DE-He213 20251030112933.0 cr nn 008mamaa 240902s2024 si | s |||| 0|eng d 9789819736591 978-981-97-3659-1 10.1007/978-981-97-3659-1 doi QA319-329.9 PBKF bicssc MAT037000 bisacsh PBKF thema Taira, Kazuaki. author. aut http://id.loc.gov/vocabulary/relators/aut 83572 Real Analysis Methods for Markov Processes [electronic resource] : Singular Integrals and Feller Semigroups / by Kazuaki Taira. 1st ed. 2024. Singapore : Springer Nature Singapore : Imprint: Springer, 2024. XVIII, 749 p. 111 illus. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Introduction and Main Results -- Elements of Functional Analysis -- Elements of Measure Theory and Lp Spaces -- Elements of Real Analysis -- Harmonic Functions and Poisson Integrals -- Besov Spaces via Poisson Integrals -- Sobolev and Besov Spaces -- Maximum Principles in Sobolev Spaces -- Elements of Singular Integrals -- Calder´on–Zygmund Kernels and Their Commutators -- Calder´on–Zygmund Variable Kernels and Their Commutators -- Dirichlet Problems in Sobolev Spaces -- Calder´on–Zygmund Kernels and Interior Estimates -- Calder´on–Zygmund Kernels and Boundary Estimates -- Unique Solvability of the Homogeneous Dirichlet Problem -- Regular Oblique Derivative Problems in Sobolev Spaces -- Oblique Derivative Boundary Conditions -- Boundary Representation Formula for Solutions -- Boundary Regularity of Solutions -- Proof of Theorems 16.1 and 16.2 -- Markov Processes and Feller Semigroups -- Feller Semigroups with Dirichlet Condition -- Feller Semigroups with an Oblique Derivative Condition -- Feller Semigroups and Boundary Value Problems -- Feller Semigroups with a First Order Ventcel’ Boundary Condition -- Concluding Remarks. This book is devoted to real analysis methods for the problem of constructing Markov processes with boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called the Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel (Wentzell) boundary condition, on the boundary of the domain. Most likely, a Markovian particle moves both by continuous paths and by jumps in the state space and obeys the Ventcel boundary condition, which consists of six terms corresponding to diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon, and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first-order Ventcel boundary value problems for second-order elliptic Waldenfels integro-differential operators. By using the Calderón–Zygmund theory of singular integrals, we prove the existence and uniqueness of theorems in the framework of the Sobolev and Besov spaces, which extend earlier theorems due to Bony–Courrège–Priouret to the vanishing mean oscillation (VMO) case. Our proof is based on various maximum principles for second-order elliptic differential operators with discontinuous coefficients in the framework of Sobolev spaces. My approach is distinguished by the extensive use of the ideas and techniques characteristic of recent developments in the theory of singular integral operators due to Calderón and Zygmund. Moreover, we make use of an Lp variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups by me. The present book is amply illustrated; 119 figures and 12 tables are provided in such a fashion that a broad spectrum of readers understand our problem and main results. Functional analysis. Stochastic processes. 76770 Probabilities. 65724 Functional Analysis. Stochastic Processes. 76770 Probability Theory. 67428 SpringerLink (Online service) Springer Nature eBook Printed edition: 9789819736584 Printed edition: 9789819736607 Printed edition: 9789819736614 https://doi.org/10.1007/978-981-97-3659-1 ZDB-2-SMA ZDB-2-SXMS 63039 63039