Alphabeta MathAnderson reverse-time diffusion
About this page
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 15 results · all verified · 6 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
Differential Geometry
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Complex Norms and Realification
- Distributions Test Functions and Differentiation
- Fourier Transform Convolution and Approximate Identities
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- Schwartz Space and the Plancherel Theorem
- The Analytic Hahn Banach Theorem
Linear Algebra
- Complexification, Realification and Real Structures
- 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
- 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 Duality of Lᵖ and L^q
- 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
Neural Networks and Learning Dynamics
Number Theory
Numerical Pdes and Spatial Discretization
Partial Differential Equations
Probability
- Brownian Motion Construction and Continuity
- Brownian Motion, Markov Properties and Hitting Times
- Brownian Path Properties
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Conditional Distributions and Regular Conditional Probability
- Conditional Expectation
- Discrete Time Martingales
- Forward Itô SDE Well-Posedness
- Gaussian smoothing densities, scores, and endpoint limits
- Independence Borel Cantelli and Zero One Laws
- Infinite Product Measures and Kolmogorov Extension
- Itos Formula and Brownian Martingales
- Martingale Inequalities and Convergence
- Modes of Convergence for Random Variables
- Probability Spaces Random Variables and Expectation
- Stopping Times and Optional Stopping
- The Ito Integral with Respect to Brownian Motion
- Weak Convergence Tightness and Representation
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- Convex and Semicontinuous Functions on Rⁿ
- Convexity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- 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
Statistical Models and Experiments
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
Bayes-reversed transition kernels for a diffusion
Lemma
Quadratic variation of a continuous local martingale
Lemma
A translated conditional density factorizes after subtraction
Lemma
An elliptic local martingale has a Brownian driver
Lemma
A normalized parabolic Bellman equation has a bounded weak solution
Lemma
A parabolic Bellman barrier with a Lebesgue height bound
Lemma
Stopped occupation absolute continuity for uniformly elliptic Ito processes
Lemma
Forward density duality identifies an auxiliary reverse process
Lemma
Forward Kolmogorov duality gives reversed martingales
Lemma
Reversed kernels, generator, and martingale problem
Theorem
Anderson's bounded-start reverse-time diffusion theorem
Definition
Bayes-reversed Gaussian smoothing kernels
Lemma
Brownian reconstruction with a predictable elliptic coefficient
Theorem
Reverse-time SDE for an unbounded-start Markov diffusion
Lemma
Generator and martingale problem for reversed Gaussian smoothing
Theorem
Reverse Gaussian smoothing on a truncated time interval
Sources
- B. D. O. Anderson, Reverse-time diffusion equation models, Sections 3–4
- Y. Tang and H. Zhao, Score-based diffusion models via stochastic differential equations: a technical tutorial, Section 2.2
- A. W. van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 5.64, Section 5.8, pp. 80–82
- B. D. O. Anderson, Reverse-time diffusion equation models, Lemma 4
- A. W. van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 6.1 and Exercise 6.5; Theorem 7.15
- N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, AMS233
- N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, Sections 2.1 and 2.2, Theorems 2.1.9 and 2.2.4
- N. V. Krylov, On time inhomogeneous stochastic Ito equations with drift in Ld+1, Lemma 4.1
- N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, Theorem 2.2.4
- B. D. O. Anderson, Reverse-time diffusion equation models, Section 5: forward density duality
- N. V. Krylov, On time inhomogeneous stochastic Ito equations with drift in Ld+1, Lemma 4.1: occupation estimate context
- B. D. O. Anderson, Reverse-time diffusion equation models, Section 5: forward and backward Kolmogorov equations
- B. D. O. Anderson, Reverse-time diffusion equation models, Section 5
- A. W. van der Vaart, Martingales, Diffusions and Financial Mathematics, Lemma 5.77 and Theorem 7.15
- Y. Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, Sections 3.2 and Appendix A.3
- W. Tang and H. Zhao, Score-based diffusion models via stochastic differential equations: a technical tutorial, Section 2, Theorem 2.1, equation (2.3)
- M. Raginsky, Stochastic Differential Equations: A Systems-Theoretic Approach, Section 7.2