Alphabeta MathBrownian Motion Construction and Continuity
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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
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- 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
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- 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
- 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
Definition
Gaussian process
Lemma
Mean and covariance determine Gaussian finite-dimensional laws
Lemma
Positive semidefiniteness of the Brownian covariance kernel
Lemma
Consistency of Brownian finite-dimensional laws
Theorem
Kolmogorov construction of the canonical Gaussian process
Lemma
Brownian covariance is equivalent to independent stationary normal increments
Definition
Brownian motion
Theorem
Kolmogorov continuity criterion in one parameter
Lemma
Gaussian even moments for Brownian increments
Theorem
Existence of continuous Brownian motion
Corollary
Brownian paths are locally Holder below one half
Definition
Uniform-on-compacts metric on continuous path space
Lemma
Under countable choice, continuous path space is Polish
Definition
Wiener measure on continuous path space
Lemma
Borel sigma-algebra of continuous path space is generated by coordinates
Theorem
Uniqueness of Wiener measure
Theorem
Brownian scaling
Theorem
Brownian time inversion
Definition
d-dimensional Brownian motion
Corollary
Existence and scaling of d-dimensional Brownian motion
Definition
Continuous-time filtrations and all-pairs martingales
Sources
- Perla Sousi, Advanced Probability, Section 6.1
- Nobuaki Yoshida, Probability Theory, Section 6.1
- Rick Durrett, Probability: Theory and Examples, Section 7.1
- Perla Sousi, Advanced Probability, Section 6.2
- Perla Sousi, Advanced Probability, Sections 6.1-6.2
- Nobuaki Yoshida, Probability Theory, Lemma 6.1.3
- Rick Durrett, Probability: Theory and Examples, Theorem 7.1.3
- Perla Sousi, Advanced Probability, Theorem 3.19
- Nobuaki Yoshida, Probability Theory, Section 6.3
- van der Vaart and Wellner, Weak Convergence and Empirical Processes, Sections 1.3 and 1.5
- A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Section 1.3
- Perla Sousi, Advanced Probability, Section 6.3
- Nobuo Yoshida, Probability Theory, Section 6.1
- Perla Sousi, Advanced Probability, Theorem 6.7
- Nobuo Yoshida, Probability Theory, Proposition 6.1.5 and Lemmas 6.1.6--6.1.7
- Nobuo Yoshida, Probability Theory, Definition 6.1.1, Lemma 6.1.3, and Proposition 6.1.4
- Perla Sousi, Advanced Probability, Sections 6.2--6.3
- Perla Sousi, Advanced Probability, Sections 2 and 3.1