Markov processes, semigroups, and generators /
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| Author / Creator: | Kolokolʹt͡sov, V. N. (Vasiliĭ Nikitich) |
|---|---|
| Imprint: | Berlin ; New York : De Gruyter, c2011. |
| Description: | xviii, 430 p. ; 25 cm. |
| Language: | English |
| Series: | De Gruyter studies in mathematics ; 38 De Gruyter studies in mathematics 38. |
| Subject: | Markov processes. Semigroups. Group theory -- Generators. Group theory -- Generators. Markov processes. Semigroups. |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/8363406 |
Table of Contents:
- Preface
- 1.5. Stable laws as the Holtzmark distributions
- 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
- 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
- Notations
- 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
- Standard abbreviations
- 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
- I. Introduction to stochastic analysis
- 4. SDE, ¿DE 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
- 1. Tools from probability and analysis
- 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
- 1.1. Essentials of measure and probability
- 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
- 1.2. Characteristic functions
- 6.4. Local criteria for boundary points
- 6.5. Decomposable generators in R + d
- 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
- 1.3. Conditioning
- 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
- 1.4. Infinitely divisible and stable distributions
- 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
