Alphabeta MathBrownian Path Properties
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
Combinatorics
Functional Analysis
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Lebesgue Measure on Euclidean Space
- Lebesgue-Stieltjes Measures and Distribution Functions
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Number Theory
Probability
- Brownian Motion Construction and Continuity
- Brownian Motion, Markov Properties and Hitting Times
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Conditional Distributions and Regular Conditional Probability
- Conditional Expectation
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Probability Spaces Random Variables and Expectation
- Weak Convergence Tightness and Representation
- Weak Laws and Series of Independent Random Variables
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convexity
- Countability and Uncountability
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Theorem
Brownian paths are nowhere locally one-half Hölder
Theorem
Brownian paths are nowhere differentiable
Corollary
Brownian paths have infinite total variation
Definition
Quadratic variation along a partition sequence
Theorem
Brownian quadratic variation on dyadic partitions
Theorem
Uniform dyadic Brownian quadratic variation process
Corollary
Brownian one- and quadratic variation
Lemma
Brownian motion has a jointly measurable continuous version
Definition
The Brownian zero set
Lemma
The Brownian zero set has Lebesgue measure zero
Theorem
The Brownian zero set has no isolated points
Corollary
The Brownian zero set is uncountable
Lemma
Two-sided Mills bounds for the standard normal tail
Theorem
Brownian law of the iterated logarithm at infinity
Corollary
Brownian law of the iterated logarithm at zero
Corollary
The critical Hölder boundary at zero
Remark
Quadratic variation needs a partition convention
Theorem
The last Brownian zero has the arcsine law
Lemma
Brownian step-potential resolvent at zero
Theorem
Brownian positive occupation time has the arcsine law
Sources
- Nobuo Yoshida, Probability Theory, Section 6.3 (subcritical Hölder) and Section 6.4, Proposition 6.4.1 (the proved alpha > 1/2 statement)
- Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.1, Theorem 7.1.6 and the remark following it
- Rick Durrett, Probability: Theory and Examples, fifth edition, Theorem 7.1.6
- Perla Sousi, Advanced Probability, Theorem 6.41
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Section 2.8
- Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.1 (the path is not of bounded variation)
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Theorems 2.8.1-2.8.2
- Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.1 (Brownian path normalization)
- Perla Sousi, Advanced Probability, Chapter 6 (Brownian motion as a random continuous path)
- Perla Sousi, Advanced Probability, Theorem 6.39
- Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.4.1
- Rick Durrett, Probability: Theory and Examples, fifth edition, Section 7.4.1 (the zero set has measure zero)
- Perla Sousi, Advanced Probability, Theorem 6.39, printed p. 71
- Rick Durrett, Probability: Theory and Examples, fifth edition, Lemma 1.2.6 and the Gaussian tail estimates (8.5.2) in the proof of Theorem 8.5.1
- Rick Durrett, Probability: Theory and Examples, fifth edition, Theorem 8.5.1
- Rick Durrett, Probability: Theory and Examples, fifth edition, Theorem 8.5.1 with Brownian time inversion
- Nobuo Yoshida, Probability Theory, Section 6.3 (subcritical Hölder regularity and the critical-boundary remark)
- Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Section 2.8 (partition-dependence warning)
- Rick Durrett, Probability: Theory and Examples, fifth edition, Example 7.4.3 and equation (7.4.7)
- Nobuo Yoshida, Probability Theory, Lemmas 6.8.1-6.8.3, printed pp. 214-216
- Nobuo Yoshida, Probability Theory, Proposition 6.8.4, printed pp. 216-217