Wojbor A. Woyczyński

Random series and stochastic integrals : single and multiple

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.- Rosenthals 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 Doobs 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 Burkholders 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.- Burkholders 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.

Lévy Processes in the Physical Sciences

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.

Global and exploding solutions for nonlocal quadratic evolution problems

Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data.

Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

Nonlocal conservation laws of the form ut + Lu +∇ · f( u)= 0, where −L is the generator of a Levy semigroup on L 1 (R n ), are encountered in continuum mechanics as model equations with anomalous diffusion. They are generalizations of the classical Burgers equation. We study the critical case when the diffusion and nonlinear terms are balanced, e.g. L ∼ (−�) α/2 ,1 <α< 2, f( s)∼ s|s| r−1 , r = 1 + (α − 1)/n. The results include decay rates of solutions and their genuinely nonlinear asymptotic behavior as time t tends to infinity, determined by self-similar source solutions.  2001 Editions scientifiques et medicales Elsevier

Automated detection of neonate EEG sleep stages

The paper integrates and adapts a range of advanced computational, mathematical and statistical tools for the purpose of analysis of neonate sleep stages based on extensive electroencephalogram (EEG) recordings. The level of brain dysmaturity of a neonate is difficult to assess by direct physical or cognitive examination, but dysmaturity is known to be directly related to the structure of neonatal sleep as reflected in the nonstationary time series produced by EEG signals which, importantly, can be collected trough a noninvasive procedure. In the past, the assessment of sleep EEG structure has often been done manually by experienced clinicians. The goal of this paper is to develop rigorous algorithmic tools for the same purpose by providing a formal scheme to separate different sleep stages corresponding to different stationary segments of the EEG signal based on statistical analysis of the spectral and nonlinear characteristics of the sleep EEG recordings. The methods developed in this paper can, potentially, be translated to other areas of biomedical research.

Growing fractal interfaces in the presence of self-similar hopping surface diffusion

We propose and study an analytic model for growing interfaces in the presence of Brownian diffusion and hopping transport. The model is based on a continuum formulation of mass conservation at the interface, including reactions. The Burgers-KPZ equation for the rate of elevation change emerges after a number of approximations are invoked. We add to the model the possibility that surface transport may be by a hopping mechanism of a Levy flight, which leads to the (multi)fractal Burgers-KPZ model. The issue how to incorporate experimental data on the jump length distribution in our model is discussed and controlled algorithms for numerical solutions of such fractal Burgers-KPZ equations are provided.

Stochastic Models in Geosystems

This volume contains the edited proceedings of a workshop on stochastic models in geosystems held during the week of May 16, 1994 at the Institute for Mathematics and its applications at the University of Minnesota. The authors represent a broad interdisciplinary spectrum including mathematics, statistics, physics, geophysics, astrophysics, atmospheric physics, fluid mechanics, seismology and oceanography. The common underlying theme was stochastic modeling of geophysical phenomena and papers appearing in this volume reflect a number of research directions that are currently pursued in this area. From the methodological mathematical point of view most of the contributions fall within the areas of wave propagation in random media, passive scalar transport in random velocity flows, dynamical systems with random forcing and self-similarity concepts including multifractals.

Density fields in Burgers and KdV-Burgers turbulence

A model analytical description of the density field advected in a velocity field governed by the multidimensional Burgers equation is suggested. This model field satisfies the mass conservation law and, in the zero viscosity limit, coincides with the generalized solution of the continuity equation. A numerical and analytical study of the evolution of such a model density field is much more convenient than the standard method of simulation of transport of passive tracer particles in the fluid.In the 1-dimensional case, a more general Korteweg–de Vries (KdV)–Burgers equation is suggested as a model which permits an analytical treatment of the density field in a strongly nonlinear model of compressible gas which takes into account dissipative and dispersive effects as well as pressure forces, the former not being accounted for in the standard Burgers framework.The dynamical and statistical properties of the density field are studied. In particular, utilizing the above model in the 2-dimensional case and the ...

Scaling Limits of Solutions of the Heat Equation for Singular Non-Gaussian Data

Limiting distributions of the parabolically rescaled solutions of the heat equation with singular non-Gaussian initial data with long-range dependence are described in terms of their multiple stochastic integral representations.

Parameter identification for singular random fields arising in Burgers’ turbulence

Abstract The paper reports on a study of classical statistical inference problems for long-memory random fields arising as solutions of the nonlinear diffusion equation with random initial data (the Burgers’ turbulence problem).