Renming Song

Bio: Renming Song is an academic researcher from University of Illinois at Urbana–Champaign. The author has contributed to research in topics: Bounded function & Boundary (topology). The author has an hindex of 42, co-authored 267 publications receiving 6408 citations. Previous affiliations of Renming Song include University of Michigan & Nankai University.

Bernstein Functions: Theory and Applications

Abstract: Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis- often with different definitions and under different names. Among the synonyms are `Laplace exponent' instead of Bernstein function, and complete Bernstein functions are sometimes called `Pick functions', `Nevanlinna functions' or `operator monotone functions'. This monograph- now in its second revised and extended edition- offers a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.

Estimates on Green functions and Poisson kernels for symmetric stable processes

Abstract: One of the most basic and most important subfamily of Levy processes is symmetric stable processes. A symmetric α-stable process X on Rn is a Levy process whose transition density p(t , x − y) relative to the Lebesgue measure is uniquely determined by its Fourier transform ∫ Rn e ix ·ξp(t , x )dx = e−t|ξ| α . Here α must be in the interval (0, 2]. When α = 2, we get a Brownian motion running with a time clock twice as fast as the standard one. Brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics. Thus it has been intensively studied. In this paper, symmetric stable processes are referred to the case when 0 < α < 2, unless otherwise specified. In the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes. Stable processes are now widely used in physics, operations research, queuing theory, mathematical finance and risk estimation. In some physics literatures, symmetric α-stable processes are called Levy flights, and they have been applied to a wide range of very complex physics issues, such as turbulent diffusion, vortex dynamics, anomalous diffusion in rotating flows, and molecular spectral fluctuations. In mathematical finance, stable processes can be used to model stock returns in incomplete market. For these and more applications of stable processes, please see the interesting book [14] by Janicki and Weron and the references therein and the recent article [15] by Klafter, Shlesinger and Zuomofen. In order to make precise predictions about natural phenomena and to better cope with these

Potential Analysis of Stable Processes and its Extensions

Abstract: Boundary Potential Theory for Schr#x00F6 dinger Operators Based on Fractional Laplacian.- Nontangential Convergence for #x03B1 -harmonic Functions.- Eigenvalues and Eigenfunctions for Stable Processes.- Potential Theory of Subordinate Brownian Motion.

Heat kernel estimates for the Dirichlet fractional Laplacian

Abstract: In this paper, we consider the fractional Laplacian -(-?)a/2 on an open subset in Rd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C1.1 open sets. This heat kernel is also the transition density of a rotationally symmetric a-stable process killed upon leaving a C1.1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

Integral Inequalities for Convex Functions of Operators on Martingales

Abstract: Let M be a family of martingales on a probability space (Ω, A, P) and let ф be a nonnegative function on [0, ∞]. The general question underlying both [2] and the present work may be stated as follows : If U and V are operators on M with values in the set of nonnegative A measurable functions on Ω, under what further conditions does $${\lambda^{po}}{P(Vf{>}\lambda)}{\leqq}{\begin{array}{lllll} {po} \\ {po} \\ \end{array}} \,\lambda >o,f,\varepsilon M,$$ (1.1) imply \(E{\Phi}(Vf)\leqq cE{\Phi}(Uf), f \ \in \ \mathcal{M}?\)

Foundations of modern probability

Abstract: * 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

Lévy processes and infinitely divisible distributions

Abstract: 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.