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2020 | Book

Introduction and Main Results Read chapter Read first chapter

Boundary Value Problems and Markov Processes

Functional Analysis Methods for Markov Processes

Author: Prof. Kazuaki Taira

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This 3rd edition provides an insight into the mathematical crossroads formed by functional analysis (the macroscopic approach), partial differential equations (the mesoscopic approach) and probability (the microscopic approach) via the mathematics needed for the hard parts of Markov processes. It brings these three fields of analysis together, providing a comprehensive study of Markov processes from a broad perspective. The material is carefully and effectively explained, resulting in a surprisingly readable account of the subject.

The main focus is on a powerful method for future research in elliptic boundary value problems and Markov processes via semigroups, the Boutet de Monvel calculus. A broad spectrum of readers will easily appreciate the stochastic intuition that this edition conveys. In fact, the book will provide a solid foundation for both researchers and graduate students in pure and applied mathematics interested in functional analysis, partial differential equations, Markov processes and the theory of pseudo-differential operators, a modern version of the classical potential theory.

Table of Contents

Frontmatter
1. Introduction and Main Results
Abstract
This book is an easy-to-read reference providing a link among functional analysis, partial differential equations and probability. In this introductory chapter, our problems and results are stated in such a fashion that a broad spectrum of readers could understand.
Table 1.1 below gives a bird’s-eye view of Markov processes, Feller semigroups and elliptic boundary value problems and how these relate to each other.
Kazuaki Taira

Analytic and Feller Semigroups and Markov Processes

Frontmatter
2. Analytic Semigroups
Abstract
This chapter is devoted to a review of standard topics from the theory of analytic semigroups which forms a functional analytic background for the proof of Theorems 1.4 and 1.5.
Kazuaki Taira
3. Markov Processes and Feller Semigroups
Abstract
This chapter is devoted to a functional analytic approach to Markov processes in probability which forms a functional analytic background for the proof of Theorems 1.6 and 1.11.
Kazuaki Taira

Pseudo-Differential Operators and Elliptic Boundary Value Problems

Frontmatter
4. Lp Theory of Pseudo-Differential Operators
Abstract
In this chapter we present a brief description of the basic concepts and results of the L p theory of pseudo-differential operators, which may be considered as a modern version of the classical potential approach.
Especially, pseudo-differential operators provide a constructive tool to deal with existence and smoothness of solutions of partial differential equations. The theory of pseudo-differential operators continues to be one of the most influential works in modern history of analysis, and is a very refined mathematical tool whose full power is yet to be exploited.
Kazuaki Taira
5. Boutet de Monvel Calculus
Abstract
In this chapter we introduce the notion of transmission property due to Boutet de Monvel which is a condition about symbols in the normal direction at the boundary. Elliptic boundary value problems cannot be treated directly by pseudo-differential operator methods. It was Boutet de Monvel who brought in the operator-algebraic aspect with his calculus in 1971. He constructed a relatively small “algebra”, called the Boutet de Monvel algebra, which contains the boundary value problems for elliptic differential operators as well as their parametrices.
Kazuaki Taira
6. Lp Theory of Elliptic Boundary Value Problems
Abstract
In this chapter we consider the non-homogeneous general Robin problem
$$\displaystyle \begin {cases} Au = f &\mbox{in }\varOmega , \\ B{\boldsymbol \gamma }u = a(x^{\prime }) \left . \dfrac {\partial u}{\partial {\boldsymbol \nu }} + b(x^{\prime }) u\right \vert { }_{{\partial \varOmega }} = \varphi &\mbox{on }{\partial \varOmega } \end {cases} $$
under the following two conditions (H.1) and (H.2) (corresponding to conditions (A) and (B) with μ = a and γ = −b):
(H.1) a(x ′) ≥ 0 and b(x ′) ≥ 0 on ∂Ω.
(H.2) a(x ′) + b(x ′) > 0 on ∂Ω.
Here ν = −n is the unit outward normal to the boundary ∂Ω (see Figure 6.1 below).
Kazuaki Taira

Analytic Semigroups in Lp Sobolev Spaces

Frontmatter
7. Proof of Theorem 1.2
Abstract
This chapter is devoted to the proof of Theorem 1.2. The idea of our proof is stated as follows. First, we reduce the study of the boundary value problem (∗)λ to that of a first-order pseudo-differential operator \(T(\lambda ) = L\mathcal {P}(\lambda )\) on the boundary ∂D, just as in Section 4.3. Then we prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate
$$\displaystyle \Vert u\Vert _{2,p} \leq C(\lambda )\left (\left \Vert f \right \Vert { }_{p} + \left \vert \varphi \right \vert { }_{2-1/p,p} + \left \Vert u \right \Vert { }_{p} \right ). $$
More precisely, we construct a parametrix S(λ) for T(λ) in the Hörmander class \(L^{0}_{1,1/2}(\partial D)\) (Lemma 7.2), and apply the Besov-space boundedness theorem (Theorem 4.47) to S(λ) to obtain the desired estimate (1.7) (Lemma 7.1).
Kazuaki Taira
8. A Priori Estimates
Abstract
This Chapter 8 and the next Chapter 9 are devoted to the proof of Theorem 1.4. In this chapter we study the operator A p, and prove a priori estimates for the operator A p − λI (Theorem 8.3) which will play a fundamental role in the next chapter. In the proof we make good use of Agmon’s method (Proposition 8.4). This is a technique of treating a spectral parameter λ as a second-order, elliptic differential operator of an extra variable and relating the old problem to a new problem with the additional variable.
Kazuaki Taira
9. Proof of Theorem 1.4
Abstract
In this chapter we prove Theorem 1.4 (Theorems 9.1 and 9.11). Once again we make use of Agmon’s method in the proof of Theorems 9.1 and 9.11. In particular, Agmon’s method plays an important role in the proof of the surjectivity of the operator A p − λI (Proposition 9.2).
Kazuaki Taira

Waldenfels Operators, Boundary Operators and Maximum Principles

Frontmatter
10. Elliptic Waldenfels Operators and Maximum Principles
Abstract
Part IV (Chapters 10 and 11) is devoted to the general study of the maximum principles for second-order, elliptic Waldenfels operators in terms of pseudo-differential operators.
In this chapter, following Bony–Courrège–Priouret [15] we prove various maximum principles for second-order, elliptic Waldenfels operators which play an essential role throughout the book.
Kazuaki Taira
11. Boundary Operators and Boundary Maximum Principles
Abstract
Let D be a bounded domain of Euclidean space R N, with smooth boundary ∂D; its closure \(\overline {D} = D \cup {\partial D}\) is an N-dimensional, compact smooth manifold with boundary (see Figure 11.1 below). In this chapter, following Bony–Courrège–Priouret ( [15, Chapter II]) we characterize Ventcel’–Lévy boundary operators T (Theorem 11.3) and Ventcel’ boundary operators Γ = Λ + T (Theorem 11.4) defined on the compact smooth manifold \(\overline {D}\) with boundary ∂D in terms of the positive boundary maximum principle:
$$\displaystyle \,\,x^{\prime }_{0} \in {\partial D}, \,\, u \in C^{2}(\overline {D}) \hspace{.5em} \mbox{and} \hspace{.5em} u(x^{\prime }_{0}) = \max _{x \in \overline {D}} u(x) \geq 0 \Longrightarrow (\varGamma u)(x^{\prime }_{0}) \leq 0. $$
This chapter will be very useful in the study of Markov processes with general Ventcel’ boundary conditions in the last Chapter 16.
Kazuaki Taira

Feller Semigroups for Elliptic Waldenfels Operators

Frontmatter
12. Proof of Theorem 1.5 - Part (i) -
Abstract
Part V (Chapters 12 through 14) is devoted to the proof of generation theorems of Feller semigroup for second-order, elliptic Waldenfels integro-differential operators.
Kazuaki Taira
13. Proofs of Theorem 1.5, Part (ii) and Theorem 1.6
Abstract
In this chapter we prove Theorem 1.6 in Section 13.3 and part (ii) of Theorem 1.5 in Section 13.4. This chapter is the heart of the subject. In Section 13.1 general existence theorems for Feller semigroups are formulated in terms of elliptic boundary value problems with spectral parameter (Theorem 13.4). In Section 13.1 we study Feller semigroups with reflecting barrier (Theorem 13.17) and then, by using these Feller semigroups we construct Feller semigroups corresponding to such a diffusion phenomenon that either absorption or reflection phenomenon occurs at each point of the boundary (Theorem 13.22). Our proof is based on the generation theorems of Feller semigroups discussed in Chapter 3.
Kazuaki Taira
14. Proofs of Theorems 1.8, 1.9, 1.10 and 1.11
Abstract
In this chapter we prove Theorems 1.8, 1.9, 1.10 and 1.11, generalizing Theorems 1.4, 1.5 and 1.6 for second-order, elliptic Waldenfels operators.
Kazuaki Taira
15. Path Functions of Markov Processes via Semigroup Theory
Abstract
In this book we have studied mainly Markov transition functions with only informal references to the random variables which actually form the Markov processes themselves (see Section 3.3). In this chapter we study this neglected side of our subject. The discussion will have a more measure-theoretical flavor than hitherto. Section 15.1 is devoted to a review of the basic definitions and properties of Markov processes. In Section 15.2 we consider when the paths of a Markov process are actually continuous, and prove Theorem 3.19 (Corollary 15.7). In Section 15.3 we give a useful criterion for path-continuity of a Markov process {x t} in terms of the infinitesimal generator \(\mathfrak {A}\) of the associated Feller semigroup {T t} (Theorem 15.9). Section 15.4 is devoted to the study of three typical examples of multi-dimensional diffusion processes. More precisely, we prove that (1) the reflecting barrier Brownian motion (Theorem 15.11), (2) the reflecting and absorbing barrier Brownian motion (Theorem 15.14) and (3) the reflecting, absorbing and drifting barrier Brownian motion (Theorem 15.15) are multi-dimensional diffusion processes, namely, they are continuous strong Markov processes.
Kazuaki Taira

Concluding Remarks

Frontmatter
16. The State-of-the-Art of Generation Theorems for Feller Semigroups
Abstract
This book is devoted to a concise and accessible exposition of the functional analytic approach to the problem of construction of strong Markov processes with Ventcel’ boundary conditions in probability. More precisely, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space \(\overline {D} \setminus M\) until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (see Figure 16.1 below). Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators which may be considered as a modern version of the classical potential theory.
Kazuaki Taira
Backmatter
Metadata
Title
Boundary Value Problems and Markov Processes
Author
Prof. Kazuaki Taira
Copyright Year
2020
Electronic ISBN
978-3-030-48788-1
Print ISBN
978-3-030-48787-4
DOI
https://doi.org/10.1007/978-3-030-48788-1