* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

Fractional Differential Equations

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Suppose that d≥2 and α∈(1,2). Let D be a bounded C1,1 open set in Rd and b an Rd-valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for Lb=Δα/2+b⋅∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for Lb in D with explicit decay rate.

/pdf/dirichlet-heat-kernel-estimates-for-fractional-laplacian-4o2iqujcck.pdf

Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

We consider the fractional Laplacian -(-Δ) α/2 on an open subset in R d with zero exterior condition.We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

Heat Kernel Estimates for Dirichlet Fractional Laplacian

Boundary Harnack Principle for Symmetric Stable Processes

Drift transforms and Green function estimates for discontinuous processes

In this paper, we study the precise behavior of the transition density func- tions of censored (resurrected) α-stable-like processes in C 1,1 open sets in R d , where d ≥ 1 and α ∈ (1,2). We first show that the semigroup of the censored α-stable- like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.

/pdf/two-sided-heat-kernel-estimates-for-censored-stable-like-2j9mw1qfca.pdf

Renming Song

Papers

Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

Heat Kernel Estimates for Dirichlet Fractional Laplacian

Boundary Harnack Principle for Symmetric Stable Processes

Drift transforms and Green function estimates for discontinuous processes

Two-sided heat kernel estimates for censored stable-like processes