Markov Processes, Semigroups and Generators

✍ Scribed by Vassili N. Kolokoltsov

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 448
• Series: De Gruyter Studies in Mathematics; 38
• Category: Library

✦ Synopsis

Markov processes represent a universal model for a large variety of real life random evolutions. The wide flow of new ideas, tools, methods and applications constantly pours into the ever-growing stream of research on Markov processes that rapidly spreads over new fields of natural and social sciences, creating new streamlined logical paths to its turbulent boundary. Even if a given process is not Markov, it can be often inserted into a larger Markov one (Markovianization procedure) by including the key historic parameters into the state space.

This monograph gives a concise, but systematic and self-contained, exposition of the essentials of Markov processes, together with recent achievements, working from the "physical picture" - a formal pre-generator, and stressing the interplay between probabilistic (stochastic differential equations) and analytic (semigroups) tools.

The book will be useful to students and researchers. Part I can be used for a one-semester course on Brownian motion, Lévy and Markov processes, or on probabilistic methods for PDE. Part II mainly contains the author's research on Markov processes.

From the contents:

• Tools from Probability and Analysis
• Brownian motion
• Markov processes and martingales
• SDE, ψDE and martingale problems
• Processes in Euclidean spaces
• Processes in domains with a boundary
• Heat kernels for stable-like processes
• Continuous-time random walks and fractional dynamics
• Complex chains and Feynman integral

• Offers a first part on basic concepts of probability theory
• Then builds on that by conveying material on generators for stable-like processes and Levy processes
• A third part offers various applications
• Easy to follow presentation
• With examples and exercises, so useable as secondary reading for courses

✦ Table of Contents

Preface
Notations
Standard abbreviations
Contents
I Introduction to stochastic analysis
1 Tools from probability and analysis
1.1 Essentials of measure and probability
1.2 Characteristic functions
1.3 Conditioning
1.4 Infinitely divisible and stable distributions
1.5 Stable laws as the Holtzmark distributions
1.6 Unimodality of probability laws
1.7 Compactness for function spaces and measures
1.8 Fractional derivatives and pseudo-differential operators
1.9 Propagators and semigroups
2 Brownian motion (BM)
2.1 Random processes: basic notions
2.2 Definition and basic properties of BM
2.3 Construction via broken-line approximation
2.4 Construction via Hilbert-space methods
2.5 Construction via Kolmogorov’s continuity
2.6 Construction via random walks and tightness
2.7 Simplest applications of martingales
2.8 Skorohod embedding and the invariance principle
2.9 More advanced Hilbert space methods: Wiener chaos and stochastic integral
2.10 Fock spaces, Hermite polynomials and Malliavin calculus
2.11 Stationarity: OU processes and Holtzmark fields
3 Markov processes and martingales
3.1 Definition of Lévy processes
3.2 Poisson processes and integrals
3.3 Construction of Lévy processes
3.4 Subordinators
3.5 Markov processes, semigroups and propagators
3.6 Feller processes and conditionally positive operators
3.7 Diffusions and jump-type Markov processes
3.8 Markov processes on quotient spaces and reflections
3.9 Martingales
3.10 Stopping times and optional sampling
3.11 Strong Markov property; diffusions as Feller processes with continuous paths
3.12 Reflection principle and passage times
4 SDE, ΨDE and martingale problems
4.1 Markov semigroups and evolution equations
4.2 The Dirichlet problem for diffusion operators
4.3 The stationary Feynman–Kac formula
4.4 Diffusions with variable drift, Ornstein–Uhlenbeck processes
4.5 Stochastic integrals and SDE based on Lévy processes
4.6 Markov property and regularity of solutions
4.7 Stochastic integrals and quadratic variation for square-integrable martingales
4.8 Convergence of processes and semigroups
4.9 Weak convergence of martingales
4.10 Martingale problems and Markov processes
4.11 Stopping and localization
II Markov processes and beyond
5 Processes in Euclidean spaces
5.1 Direct analysis of regularity and well-posedness
5.2 Introduction to sensitivity analysis
5.3 The Lie–Trotter type limits and T -products
5.4 Martingale problems for Lévy type generators: existence
5.5 Martingale problems for Lévy type generators: moments
5.6 Martingale problems for Lévy type generators: unbounded coefficients
5.7 Decomposable generators
5.8 SDEs driven by nonlinear Lévy noise
5.9 Stochastic monotonicity and duality
5.10 Stochastic scattering
5.11 Nonlinear Markov chains, interacting particles and deterministic processes
5.12 Comments
6 Processes in domains with a boundary
6.1 Stopped processes and boundary points
6.2 Dirichlet problem and mixed initial-boundary problem
6.3 The method of Lyapunov functions
6.4 Local criteria for boundary points
6.5 Decomposable generators in RdC
6.6 Gluing boundary
6.7 Processes on the half-line
6.8 Generators of reflected processes
6.9 Application to interacting particles: stochastic LLN
6.10 Application to evolutionary games
6.11 Application to finances: barrier options, credit derivatives, etc.
6.12 Comments
7 Heat kernels for stable-like processes
7.1 One-dimensional stable laws: asymptotic expansions
7.2 Stable laws: asymptotic expansions and identities
7.3 Stable laws: bounds
7.4 Stable laws: auxiliary convolution estimates
7.5 Stable-like processes: heat kernel estimates
7.6 Stable-like processes: Feller property
7.7 Application to sample-path properties
7.8 Application to stochastic control
7.9 Application to Langevin equations driven by a stable noise
7.10 Comments
8 CTRW and fractional dynamics
8.1 Convergence of Markov semigroups and processes
8.2 Diffusive approximations for random walks and CLT
8.3 Stable-like limits for position-dependent random walks
8.4 Subordination by hitting times and generalized fractional evolutions
8.5 Limit theorems for position dependent CTRW
8.6 Comments
9 Complex Markov chains and Feynman integral
9.1 Infinitely-divisible complex distributions and complex Markov chains
9.2 Path integral and perturbation theory
9.3 Extensions
9.4 Regularization of the Schrödinger equation by complex time or mass, or continuous observation
9.5 Singular and growing potentials, magnetic fields and curvilinear state spaces
9.6 Fock-space representation
9.7 Comments
Bibliography
Index