This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.
After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô’s fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.
Author(s): Rabi Bhattacharya , Edward C. Waymire
Series: Graduate Texts in Mathematics 299
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 506
City: Cham
Tags: Hille-Yoshida theorem, Lévy processes, stochastic integral, Stochastic differential equations, Markov processes with jumps, Markov property, central limit theorem, infinitely divisible distributions, jump phenomena, semigroups
Preface
A Trilogy on Stochastic Processes
Contents
1 A Review of Martingales, Stopping Times, and the Markov Property
Exercises
2 Semigroup Theory and Markov Processes
Exercises
3 Regularity of Markov Process Sample Paths
Exercises
4 Continuous Parameter Jump Markov Processes
Exercises
5 Processes with Independent Increments
Exercises
6 The Stochastic Integral
Exercises
7 Construction of Diffusions as Solutions of Stochastic Differential Equations
7.1 Construction of One-Dimensional Diffusions
7.2 Extension to Multidimensional Diffusions
7.3 An Extension of the Itô Integral & SDEs with Locally Lipschitz Coefficients
7.4 Strong Markov Property
7.5 An Extension to SDEs with Nonhomogeneous Coefficients
7.6 An Extension to k-Dimensional SDE Governed by r-Dimensional Brownian Motion
Exercises
8 Itô's Lemma
8.1 Asymptotic Properties of One-Dimensional Diffusions: Transience and Recurrence
Exercises
9 Cameron–Martin–Girsanov Theorem
Exercises
10 Support of Nonsingular Diffusions
Exercises
11 Transience and Recurrence of Multidimensional Diffusions
Exercises
12 Criteria for Explosion
Exercises
13 Absorption, Reflection, and Other Transformations of Markov Processes
13.1 Absorption
13.2 General One-Dimensional Diffusions on Half-Line with Absorption at Zero
13.3 Reflecting Diffusions
Exercises
14 The Speed of Convergence to Equilibrium of Discrete Parameter Markov Processes and Diffusions
Exercises
15 Probabilistic Representation of Solutions to Certain PDEs
15.1 Feynman–Kaĉ Formula for Multidimensional Diffusion
15.2 Kolmogorov Forward Equation (The Fokker–Planck Equation)
Exercises
16 Probabilistic Solution of the Classical Dirichlet Problem
Exercises
17 The Functional Central Limit Theorem for Ergodic Markov Processes
17.1 A Functional Central Limit Theorem for Diffusions with Periodic Coefficients
Exercises
18 Asymptotic Stability for Singular Diffusions
Exercises
19 Stochastic Integrals with L2-Martingales
Exercises
20 Local Time for Brownian Motion
Exercises
21 Construction of One-Dimensional Diffusions by Semigroups
Exercises
22 Eigenfunction Expansions of Transition Probabilities for One-Dimensional Diffusions
Exercises
23 Special Topic: The Martingale Problem
Exercises
24 Special Topic: Multiphase Homogenization for Transport in Periodic Media
Exercises
25 Special Topic: Skew Random Walk and Skew Brownian Motion
Exercises
26 Special Topic: Piecewise Deterministic Markov Processes in Population Biology
Exercises
A The Hille–Yosida Theorem and Closed Graph Theorem
References
Related Textbooks and Monographs
Symbol Index
Author Index
Subject Index
