Author
Wojbor A. Woyczyński
Other affiliations: Polish Academy of Sciences, Michigan State University
Bio: Wojbor A. Woyczyński is an academic researcher from Case Western Reserve University. The author has contributed to research in topics: Nonlinear system & Burgers' equation. The author has an hindex of 31, co-authored 126 publications receiving 3275 citations. Previous affiliations of Wojbor A. Woyczyński include Polish Academy of Sciences & Michigan State University.
Papers published on a yearly basis
Papers
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Book•
01 Jul 1992
TL;DR: In this article, the convergence of the p-th mean of a series of tangent sequences has been studied in the context of Gaussian random series, and the convergence has been shown to be a strong predictor of strong domination of character systems on Abelian groups.
Abstract: 0 Preliminaries.- 0.1 Topology and measures.- 0.2 Tail inequalities.- 0.3 Filtrations and stopping times.- 0.4 Extensions of probability spaces.- 0.5 Bernoulli and canonical Gaussian and ?-stable sequences.- 0.6 Gaussian measures on linear spaces.- 0.7 Modulars on linear spaces.- 0.8 Musielak-Orlicz spaces.- 0.9 Random Musielak-Orlicz spaces.- 0.10 Complements and comments.- Bibliographical notes.- I Random Series.- 1 Basic Inequalities for Random Linear Forms in Independent Random Variables.- 1.1 Levy-Octaviani inequalities.- 1.2 Contraction inequalities.- 1.3 Moment inequalities.- 1.4 Complements and comments.- Best constants in the Levy-Octaviani inequality.- A contraction inequality for mixtures of Gaussian random variables.- Tail inequalities for Bernoulli and Gaussian random linear forms.- A refinement of the moment inequality.- Comparison of moments.- Bibliographical notes.- 2 Convergence of Series of Independent Random Variables.- 2.1 The Ito-Nisio Theorem.- 2.2 Convergence in the p-th mean.- 2.3 Exponential and other moments of random series.- 2.4 Random series in function spaces.- 2.5 An example: construction of the Brownian motion.- 2.6 Karhunen-Loeve representation of Gaussian measures.- 2.7 Complements and comments.- Rosenthal's inequalities.- Strong exponential moments of Gaussian series.- Lattice function spaces.- Convergence of Gaussian series.- Bibliographical notes.- 3 Domination Principles and Comparison of Sums of Independent Random Variables.- 3.1 Weak domination.- 3.2 Strong domination.- 3.3 Hypercontractive domination.- 3.4 Hypercontractivity of Bernoulli and Gaussian series.- 3.5 Sharp estimates of growth of p-th moments.- 3.6 Complements and comments.- More on C-domination.- Superstrong domination.- Domination of character systems on compact Abelian groups.- Random matrices.- Hypercontractivity of real random variables.- More precise estimates on strong exponential moments of Gaussian series.- Growth of p-th moments revisited.- More on strong exponential moments of series of bounded variables.- Bibliographical notes.- 4 Martingales.- 4.1 Doob's inequalities.- 4.2 Convergence of martingales.- 4.3 Tangent and decoupled sequences.- 4.4 Complements and comments.- Bibliographical notes.- 5 Domination Principles for Martingales.- 5.1 Weak domination.- 5.2 Strong domination.- 5.3 Burkholder's method: comparison of subordinated martingales.- 5.4 Comparison of strongly dominated martingales.- 5.5 Gaussian martingales.- 5.6 Classic martingale inequalities.- 5.7 Comparison of the a.s convergence of series of tangent sequences.- 5.8 Complements and comments.- Tangency and ergodic theorems.- Burkholder's method for conditionally Gaussian and conditionally independent martingales.- Necessity of moderate growth of ?.- Comparison of Gaussian martingales revisited.- Comparing H-valued martingales with 2-D martingales.- The principle of conditioning in limit theorems.- Bibliographical notes.- 6 Random Multilinear Forms in Independent Random Variables and Polynomial Chaos.- 6.1 Basic definitions and properties.- 6.2 Maximal inequalities.- 6.3 Contraction inequalities and domination of polynomial chaos.- 6.4 Decoupling inequalities.- 6.5 Comparison of moments of polynomial chaos.- 6.6 Convergence of polynomial chaos.- 6.7 Quadratic chaos.- 6.8 Wiener chaos and Herrnite polynomials.- 6.9 Complements and comments.- Tail and moment comparisons for chaos and its decoupled chaos.- Necessity of the symmetry condition in decoupling inequalities.- Karhunen-Loeve expansion for the Wiener chaos.- ?-stable chaos of degree d ? 2.- Bibliographical notes.- II Stochastic Integrals.- 7 Integration with Respect to General Stochastic Measures.- 7.1 Construction of the integral.- 7.2 Examples of stochastic measures.- 7.3 Complements and comments.- An alternative definition of m-integrability.- Bibliographical notes.- 8 Deterministic Integrands.- 8.1 Discrete stochastic measure.- 8.2 Processes with independent increments and their characteristics.- 8.3 Integration with respect to a general independently scattered measure.- 8.4 Complements and comments.- Stochastic measures with finite p-th moments.- Bibliographical notes.- 9 Predictable Integrands.- 9.1 Integration with respect to processes with independent increments: Decoupling inequalities approach.- 9.2 Brownian integrals.- 9.3 Characteristics of semimartingales.- 9.4 Semimartingale integrals.- 9.5 Complements and comments.- The Bichteler-Dellacherie Theorem.- Semimartingale integrals in Lp.- ?-stable integrals.- Bibliographical notes.- 10 Multiple Stochastic Integrals.- 10.1 Products of stochastic measures.- 10.2 Structure of double integrals.- 10.3 Wiener polynomial chaos revisited.- 10.4 Complements and comments.- Multiple ?-stable integrals.- Bibliographical notes.- A Unconditional and Bounded Multiplier Convergence of Random Series.- A.2 Almost sure convergence.- A.3 Complements and comments.- A hypercontractive view.- Bibliographical notes.- B Vector Measures.- B.1 Extensions of vector measures.- B.2 Boundedness and control measure of stochastic measures.- B.3 Complements and comments.- Bibliographical notes.
469 citations
TL;DR: In this paper, the authors studied local and global in time solutions to a class of generalized Burgers-type equations with a fractional power of the Laplacian in the principal part and with general algebraic nonlinearity.
215 citations
TL;DR: In this paper, a model for anomalous diffusion processes with random jump sizes and random inter-jump time intervals with infinite means (and variances) which do not satisfy the Law of Large Numbers is discussed.
Abstract: The paper discusses a model for anomalous diffusion processes. Their one-point probability density functions (p.d.f.) are exact solutions of fractional diffusion equations. The model reflects the asymptotic behavior of a jump (anomalous random walk) process with random jump sizes and random inter-jump time intervals with infinite means (and variances) which do not satisfy the Law of Large Numbers. In the case when these intervals have a fractional exponential p.d.f., the fractional Komogorov–Feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed. Finally, some statistical properties of solutions of the related Langevin equation are studied. The subdiffusive case is explored in detail. The emphasis is on a rigorous presentation which, however, would be accessible to the physical sciences audience.
184 citations
01 Jan 2001
TL;DR: In this article, the authors review a number of physical phenomena for which the Levy processes and α-stable processes can be used as a reasonable model and discuss the asymptotic and approximation schemes.
Abstract: We review a number of physical phenomena for which the Levy processes and, in particular, α-stable processes can be used as a reasonable model. Examples from fluid mechanics, solid state physics, polymer chemistry, and mathematical finance leading to such non-Gaussian processes are described. For nonlinear problems, the asymptotic and approximation schemes are discussed. The article is written with both mathematical and physical sciences audiences in mind.
164 citations
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Book•
01 Jan 1997
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
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
4,562 citations
TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
Abstract: This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
3,555 citations
TL;DR: In this article, Modelling Extremal Events for Insurance and Finance is discussed. But the authors focus on the modeling of extreme events for insurance and finance, and do not consider the effects of cyber-attacks.
Abstract: (2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.
2,729 citations
2,345 citations
Book•
01 Jan 2001TL;DR: Concentration functions and inequalities isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index
Abstract: Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index
2,324 citations