Bessel process

About: Bessel process is a research topic. Over the lifetime, 1521 publications have been published within this topic receiving 40651 citations.

A treatise on the theory of Bessel functions

Abstract: 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.

Continuous martingales and Brownian motion

Abstract: 0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov's Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- 1. Gronwall's Lemma.- 2. Distributions.- 3. Convex Functions.- 4. Hausdorff Measures and Dimension.- 5. Ergodic Theory.- 6. Probabilities on Function Spaces.- 7. Bessel Functions.- 8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.

Diffusion Processes and their Sample Paths

Abstract: Prerequisites.- 1. The standard BRownian motion.- 1.1. The standard random walk.- 1.2. Passage times for the standard random walk.- 1.3. Hin?in's proof of the de Moivre-laplace limit theorem.- 1.4. The standard Brownian motion.- 1.5. P. Levy's construction.- 1.6. Strict Markov character.- 1.7. Passage times for the standard Brownian motion.- Note l: Homogeneous differential processes with increasing paths.- 1.8. Kolmogorov's test and the law of the iterated logarithm.- 1.9. P. Levy's Holder condition.- 1.10. Approximating the Brownian motion by a random walk.- 2. Brownian local times.- 2.1. The reflecting Brownian motion.- 2.2. P. Levy's local time.- 2.3. Elastic Brownian motion.- 2.4. t+ and down-crossings.- 2.5. T+ as Hausdorff-Besicovitch 1/2-dimensional measure.- Note 1: Submartingales.- Note 2: Hausdorff measure and dimension.- 2.6. Kac's formula for Brownian functionals.- 2.7. Bessel processes.- 2.8. Standard Brownian local time.- 2.9. BrowNian excursions.- 2.10. Application of the Bessel process to Brownian excursions.- 2.11. A time substitution.- 3. The general 1-dimensional diffusion.- 3.1. Definition.- 3.2. Markov times.- 3.3. Matching numbers.- 3.4. Singular points.- 3.5. Decomposing the general diffusion into simple pieces.- 3.6. Green operators and the space D.- 3.7. Generators.- 3.8. Generators continued.- 3.9. Stopped diffusion.- 4. Generators.- 4.1. A general view.- 4.2. G as local differential operator: conservative non-singular case.- 4.3. G as local differential operator: general non-singular case.- 4.4. A second proof.- 4.5. G at an isolated singular point.- 4.6. Solving G*u = ? u.- 4.7. G as global differential operator: non-singular case.- 4.8. G on the shunts.- 4.9. G as global differential operator: singular case.- 4.10. Passage times.- Note 1: Differential processes with increasing paths.- 4.11. Eigen-differential expansions for Green functions and transition densities.- 4.12. Kolmogorov's test.- 5. Time changes and killing.- 5.1. Construction of sample paths: a general view.- 5.2. Time changes: Q = R1.- 5.3. Time changes: Q = [0, + ?).- 5.4. Local times.- 5.5. Subordination and chain rule.- 5.6. Killing times.- 5.7. Feller's Brownian motions.- 5.8. Ikeda's example.- 5.9. Time substitutions must come from local time integrals.- 5.10. Shunts.- 5.11. Shunts with killing.- 5.12. Creation of mass.- 5.13. A parabolic equation.- 5.14. Explosions.- 5.15. A non-linear parabolic equation.- 6. Local and inverse local times.- 6.1. Local and inverse local times.- 6.2. Levy measures.- 6.3. t and the intervals of [0, + ?) - ?.- 6.4. A counter example: t and the intervals of [0, + ?) - ?.- 6.5a t and downcrossings.- 6.5b t as Hausdorff measure.- 6.5c t as diffusion.- 6.5d Excursions.- 6.6. Dimension numbers.- 6.7. Comparison tests.- Note 1: Dimension numbers and fractional dimensional capacities.- 6.8. An individual ergodic theorem.- 7. Brownian motion in several dimensions.- 7.1. Diffusion in several dimensions.- 7.2. The standard Brownian motion in several dimensions.- 7.3. Wandering out to ?.- 7.4. Greenian domains and Green functions.- 7.5. Excessive functions.- 7.6. Application to the spectrum of ?/2.- 7.7. Potentials and hitting probabilities.- 7.8. Newtonian capacities.- 7.9. Gauss's quadratic form.- 7.10. Wiener's test.- 7.11. Applications of Wiener's test.- 7.12. Dirichlet problem.- 7.13. Neumann problem.- 7.14. Space-time Brownian motion.- 7.15. Spherical Brownian motion and skew products.- 7.16. Spinning.- 7.17. An individual ergodic theorem for the standard 2-dimensional BROWNian motion.- 7.18. Covering Brownian motions.- 7.19. Diffusions with Brownian hitting probabilities.- 7.20. Right-continuous paths.- 7.21. Riesz potentials.- 8. A general view of diffusion in several dimensions.- 8.1. Similar diffusions.- 8.2. G as differential operator.- 8.3. Time substitutions.- 8.4. Potentials.- 8.5. Boundaries.- 8.6. Elliptic operators.- 8.7. Feller's little boundary and tail algebras.- List of notations.

Mathematical Methods in the Physical Sciences

Abstract: Chapter 1: Infinite Series, Power Series.The Geometric Series.Definitions and Notation.Applications of Series.Convergent and Divergent Series.Convergence Tests.Convergence Tests for Series of Positive Terms.Alternating Series.Conditionally Convergent Series.Useful Facts about Series.Power Series Interval of Convergence.Theorems about Power Series.Expanding Functions in Power Series.Expansion Techniques.Accuracy of Series Approximations.Some Uses of Series.Chapter 2: Complex Numbers.Introduction.Real and Imaginary Parts of a Complex Number.The Complex Plane.Terminology and Notation.Complex Algebra.Complex Infinite Series.Complex Power Series Disk of Convergence.Elementary Functions of Complex Numbers.Euler's Formula.Powers and Roots of Complex Numbers.The Exponential and Trigonometric Functions.Hyperbolic Functions.Logarithms.Complex Roots and Powers.Inverse Trigonometric and Hyperbolic Functions.Some Applications.Chapter 3: Linear Algebra.Introduction.Matrices Row Reduction.Determinants Cramer's Rule.Vectors.Lines and Planes.Matrix Operations.Linear Combinations, Functions, Operators.Linear Dependence and Independence.Special Matrices and Formulas.Linear Vector Spaces.Eigenvalues and Eigenvectors.Applications of Diagonalization.A Brief Introduction to Groups.General Vector Spaces.Chapter 4: Partial Differentiation.Introduction and Notation.Power Series in Two Variables.Total Differentials.Approximations using Differentials.Chain Rule.Implicit Differentiation.More Chain Rule.Maximum and Minimum Problems.Constraints Lagrange Multipliers.Endpoint or Boundary Point Problems.Change of Variables.Differentiation of Integrals.Chapter 5: Multiple Integrals.Introduction.Double and Triple Integrals.Applications of Integration.Change of Variables in Integrals Jacobians.Surface Integrals.Chapter 6: Vector Analysis.Introduction.Applications of Vector Multiplication.Triple Products.Differentiation of Vectors.Fields.Directional Derivative Gradient.Some Other Expressions Involving V.Line Integrals.Green's Theorems in the Plane.The Divergence and the Divergence Theorem.The Curl and Stokes' Theorem.Chapter 7: Fourier Series and Transforms.Introduction.Simple Harmonic Motion and Wave Motion Periodic Functions.Applications of Fourier Series.Average Value of a Function.Fourier Coefficients.Complex Form of Fourier Series.Other Intervals.Even and Odd Functions.An Application to Sound.Parseval's Theorem.Fourier Transforms.Chapter 8: Ordinary Differential Equations.Introduction.Separable Equations.Linear First-Order Equations.Other Methods for First-Order Equations.Linear Equations (Zero Right-Hand Side).Linear Equations (Nonzero Right-Hand Side).Other Second-Order Equations.The Laplace Transform.Laplace Transform Solutions.Convolution.The Dirac Delta Function.A Brief Introduction to Green's Functions.Chapter 9: Calculus of Variations.Introduction.The Euler Equation.Using the Euler Equation.The Brachistochrone Problem Cycloids.Several Dependent Variables Lagrange's Equations.Isoperimetric Problems.Variational Notation.Chapter 10: Tensor Analysis.Introduction.Cartesian Tensors.Tensor Notation and Operations.Inertia Tensor.Kronecker Delta and Levi-Civita Symbol.Pseudovectors and Pseudotensors.More about Applications.Curvilinear Coordinates.Vector Operators.Non-Cartesian Tensors.Chapter 11: Special Functions.Introduction.The Factorial Function.Gamma Function Recursion Relation.The Gamma Function of Negative Numbers.Formulas Involving Gamma Functions.Beta Functions.Beta Functions in Terms of Gamma Functions.The Simple Pendulum.The Error Function.Asymptotic Series.Stirling's Formula.Elliptic Integrals and Functions.Chapter 12: Legendre, Bessel, Hermite, and Laguerre functions.Introduction.Legendre's Equation.Leibniz' Rule for Differentiating Products.Rodrigues' Formula.Generating Function for Legendre Polynomials.Complete Sets of Orthogonal Functions.Orthogonality of Legendre Polynomials.Normalization of Legendre Polynomials.Legendre Series.The Associated Legendre Polynomials.Generalized Power Series or the Method of Frobenius.Bessel's Equation.The Second Solutions of Bessel's Equation.Graphs and Zeros of Bessel Functions.Recursion Relations.Differential Equations with Bessel Function Solutions.Other Kinds of Bessel Functions.The Lengthening Pendulum.Orthogonality of Bessel Functions.Approximate Formulas of Bessel Functions.Series Solutions Fuch's Theorem.Hermite and Laguerre Functions Ladder Operators.Chapter 13: Partial Differential Equations.Introduction.Laplace's Equation Steady-State Temperature.The Diffusion of Heat Flow Equation the Schrodinger Equation.The Wave Equation the Vibrating String.Steady-State Temperature in a Cylinder.Vibration of a Circular Membrane.Steady-State Temperature in a Sphere.Poisson's Equation.Integral Transform Solutions of Partial Differential Equations.Chapter 14: Functions of a Complex Variable.Introduction.Analytic Functions.Contour Integrals.Laurent Series.The Residue Theorem.Methods of Finding Residues.Evaluation of Definite Integrals.The Point at Infinity Residues of Infinity.Mapping.Some Applications of Conformal Mapping.Chapter 15: Probability and Statistics.Introduction.Sample Space.Probability Theorems.Methods of Counting.Random Variables.Continuous Distributions.Binomial Distribution.The Normal or Gaussian Distribution.The Poisson Distribution.Statistics and Experimental Measurements.