Series: Lecture Notes in Mathematics , Vol. 1499
2009, Approx. 200 p., Softcover
ISBN: 978-3-642-01676-9
Due: July 16, 2009
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.
1 Semigroup Theory.- 2 Lp Theory of Pseudo-Differential Operators.- 3 Lp Approach to Elliptic Boundary Value Problems.- 4 Proof of Theorem 1.- 5 A Priori Estimates.- 6 Proof of Theorem 2.- 7 Proof of Theorem 3, Part (i) .- 8 Proof of Theorem 3, Part (ii).- 9 Application to Semilinear Initial-Boundary Value Problems.
Series: Lecture Notes in Mathematics , Vol. 1978
2009, Approx. 290 p., Softcover
SBN: 978-3-642-01953-1
Due: July 2009
This volume presents a general smooth ergodic theory for deterministic dynamical systems generated by non-invertible endomorphisms, mainly concerning the relations between entropy, Lyapunov exponents and dimensions.
The authors make extensive use of the combination of the inverse limit space technique and the techniques developed to tackle random dynamical systems. The most interesting results in this book are (1) the equivalence between the SRB property and Pesinfs entropy formula; (2) the generalized Ledrappier-Young entropy formula; (3) exact-dimensionality for weakly hyperbolic diffeomorphisms and for expanding maps. The proof of the exact-dimensionality for weakly hyperbolic diffeomorphisms seems more accessible than that of Barreira et al. It also inspires the authors to argue to what extent the famous Eckmann-Ruelle conjecture and many other classical results for diffeomorphisms and for flows hold true.
After a careful reading of the book, one can systematically learn the Pesin theory for endomorphisms as well as the typical tricks played in the estimation of the number of balls of certain properties, which are extensively used in Chapters IX and X.
I Preliminaries.- II Margulis-Ruelle Inequality.- III Expanding Maps.- IV Axiom A Endomorphisms.- V Unstable and Stable Manifolds.- VI Pesinfs Entropy Formula.- VII SRB Measures and Entropy Formula.- VIII Ergodic Property of Lyapunov Exponents.- IX Generalized Entropy Formula.- X Dimension of Hyperbolic Measures.
Series: Lecture Notes in Mathematics , Vol. 1980
2009, Approx. 190 p., Softcover
ISBN: 978-3-642-02140-4
Due: July 2009
Stable Levy processes and related stochastic processes play an important role in stochastic modelling in applied sciences, in particular in financial mathematics. This book is about the potential theory of stable stochastic processes. It also deals with related topics, such as the subordinate Brownian motions (including the relativistic process) and Feynman?Kac semigroups generated by certain Schroedinger operators. The authors focus on classes of stable and related processes that contain the Brownian motion as a special case.
This is the first book devoted to the probabilistic potential theory of stable stochastic processes, and, from the analytical point of view, of the fractional Laplacian. The introduction is accessible to non-specialists and provides a general presentation of the fundamental objects of the theory. Besides recent and deep scientific results the book also provides a didactic approach to its topic, as all chapters have been tested on a wide audience, including young mathematicians at a CNRS/HARP Workshop, Angers 2006.
The reader will gain insight into the modern theory of stable and related processes and their potential analysis with a theoretical motivation for the study of their fine properties.
1 Introduction.- 2 Boundary Potential Theory.- 3 Nontangential Convergence.- 4 Eigenvalues and Eigenfunctions for Stable Processes.- 5 Potential Theory of Subordinate Brownian Motion.
Series: Lecture Notes in Mathematics , Vol. 1982
2009, Approx. 300 p., Softcover
ISBN: 978-3-642-02379-8
Due: July 2009
This volume gives a unified presentation of stochastic analysis for continuous and discontinuous stochastic processes, in both discrete and continuous time. It is mostly self-contained and accessible to graduate students and researchers having already received a basic training in probability. The simultaneous treatment of continuous and jump processes is done in the framework of normal martingales; that includes the Brownian motion and compensated Poisson processes as specific cases. In particular, the basic tools of stochastic analysis (chaos representation, gradient, divergence, integration by parts) are presented in this general setting. Applications are given to functional and deviation inequalities and mathematical finance.
1 The Discrete Time Case.- 2 Continuous Time Normal Martingales.- 3 Gradient and Divergence Operators.- 4 Annihilation and creation operators.- 5 Analysis on the Wiener Space.- 6 Analysis on the Poisson space.- 7 Local Gradients on the Poisson space.- 8 Option Hedging in Continuous Time