* 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

http://www.math.pku.edu.cn/teachers/renyx/Publication/39

Central limit theorems for supercritical branching Markov processes

A fruitful analogy in harmonic analysis is the analogy between a conjugate harmonic function and a martingale transform. One idea that underlies both of these concepts is the idea of differential subordination. Our study of it here yields new information about harmonic functions and martingales, and their interaction.

Differential Subordination of Harmonic Functions and Martingales

Let $J$ be the Levy density of a symmetric Levy process in $\mathbb{R}^d$ with its Levy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ {\mathcal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>\epsilon\}}(f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$ where $\kappa(x,z)$ is a Borel measurable function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and $|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$. We construct the heat kernel $p^\kappa(t, x, y)$ of ${\mathcal L}^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$.

Heat kernels of non-symmetric jump processes: beyond the stable case

Let $\wh{; ; ; X}; ; ; =C-Y$ where $Y$ is a general one-dimensional Levy process and $C$ an independent subordinator. Consider the times when a new supremum of $\wh{; ; ; X}; ; ; $ is reached by a jump of the subordinator $C$. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and $\wh{; ; ; X}; ; ; $ drifts to $-\infty$, we decompose the absolute supremum of $\wh{; ; ; X}; ; ; $ at these times, and derive a Pollaczek-Hinchin-type formula for the distribution function of the supremum.

/pdf/on-suprema-of-levy-processes-and-application-in-risk-theory-29vfbsfrsz.pdf

On suprema of Lévy processes and application in risk theory

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L + μ ∇ ¯ ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ = (μ 1 ,... μ d ) is such that each component μ i , i = 1,..., d, is a signed measure belonging to the Kato class K d , and v is a (nonnegative) measure belonging to the Kato class K d,2 . We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for Y. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion Y D with measure-valued drift and potential when D is one of the following types of bounded domains: twisted Holder domains of order a e (1/3, 1], uniformly Holder domains of order a ∈ (0, 2) and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181-206] and [Probab. Theory Related Fields 91 (1992) 405-443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of Y D is finite.

/pdf/intrinsic-ultracontractivity-of-nonsymmetric-diffusions-with-2kwn7gaei1.pdf

Renming Song

Papers

Central limit theorems for supercritical branching Markov processes

Differential Subordination of Harmonic Functions and Martingales

Heat kernels of non-symmetric jump processes: beyond the stable case

On suprema of Lévy processes and application in risk theory

Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials