* 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

In this paper, we study the asymptotic behavior, as the time t goes to zero, of the trace of the semigroup of a killed relativistic α-stable process in bounded C 1,1 open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of t of the trace with an error bound of order t 2/α t −d/α for C 1,1 open sets and of order t 1/α t −d/α for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders t −d/α and t (2−d)/α for C 1,1 open sets, and, when α ∈ (0, 1), between the orders t −d/α and t (1−d)/α for Lipschitz open sets.

Trace Estimates for Relativistic Stable Processes

Let $X$ be a standard Markov process with state space $E$ and let $F$ be a closed subset of $E$. A nonnegative function $f$ on $F$ is extended probabilistically to a function $h_f$ on the whole space $E$. We show that the extension $h_f$ is harmonic with respect to $X$ provided that $f$ is harmonic with respect to $Y$, the trace process on $F$ of the process $X$. A consequence is that if the Harnack inequality holds for $X$, it also holds for the trace process $Y$. Several examples illustrating the usefulness of the result are given.

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On harmonic functions for trace processes

A subordinator is called special if the restriction of its potential measure to (0,∞) has a decreasing density with respect to Lebesgue measure. In this note we investigate what type of measures μ on (0,∞) can arise as Levy measures of special subordinators and what type of functions u : (0,∞)→ [0,∞) can arise as potential densities of special subordinators. As an application of the main result, we give examples of potential densities of subordinators which are constant to the right of a positive number. AMS 2000 Mathematics Subject Classification: Primary 60G51, Secondary 60J45, 60J75.

Some Remarks on Special Subordinators

Recently in [17, 18], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups and proved that for a large class of non-symmetric difiusions Z with measure-valued drift and potential, the semigroup of Z D (the process obtained by killing Z upon exiting D) in a bounded domain is intrinsic ultracontractive under very mild assumptions. In this paper, we study the intrinsic ultracontractivity for non-symmetric discontin

/pdf/intrinsic-ultracontractivity-for-non-symmetric-levy-11oqhifghk.pdf

Intrinsic Ultracontractivity for Non-symmetric Levy Processes

Recently we extended the concept of intrinsic ultracontractivity to non-symmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of Z^D (the process obtained by killing Z upon exiting D) in a bounded domain is intrinsic ultracontractive under very mild assumptions. 
In this paper, we study the intrinsic ultracontractivity for non-symmetric discontinuous Levy processes. We prove that, for a large class of non-symmetric discontinuous Levy processes X such that the Lebesgue measure is absolutely continuous with respect to the Levy measure of X, the semigroup of X^D in any bounded open set D is intrinsic ultracontractive. In particular, for the non-symmetric stable process X, the semigroup of X^D is intrinsic ultracontractive for any bounded set D. Using the intrinsic ultracontractivity, we show that the parabolic boundary Harnack principle is true for those processes. Moreover, we get that the supremum of the expected conditional lifetimes in a bounded open set is finite. We also have results of the same nature when the Levy measure is compactly supported.

Renming Song

Papers

Trace Estimates for Relativistic Stable Processes

On harmonic functions for trace processes

Some Remarks on Special Subordinators

Intrinsic Ultracontractivity for Non-symmetric Levy Processes

Intrinsic Ultracontractivity for Non-symmetric Levy Processes