LeanLevy

A Lean 4 formalization of Lévy processes, built on top of mathlib.

What's here

Fourier analysis (LeanLevy/Fourier/)

• Fourier transform of finite measures on ℝ, with boundedness and continuity

Characteristic functions (LeanLevy/Probability/Characteristic.lean)

• Characteristic function of probability measures
• Bochner positive semi-definiteness
• Multiplicativity under convolution

Poisson distribution (LeanLevy/Probability/Poisson.lean)

• Expectation, variance, second factorial moment
• Characteristic function: φ(ξ) = exp(r(exp(iξ) − 1))

Lévy's continuity theorem (LeanLevy/Probability/WeakConvergence.lean)

• Both directions fully proved: weak convergence ⟺ pointwise convergence of characteristic functions
• Tightness from characteristic function convergence

Stochastic processes (LeanLevy/Processes/StochasticProcess.lean)

• Independent and stationary increments
• Increment independence from the natural filtration

Finite-dimensional distributions (LeanLevy/Processes/FiniteDimensional.lean)

• Joint law at finitely many times as a pushforward measure
• Marginalization: restricting to a subset of times recovers the sub-distribution
• Projective consistency (IsProjectiveMeasureFamily)

Projective families (LeanLevy/Processes/ProjectiveFamily.lean)

• Bundled structure: measure family + consistency + probability
• Projection and composition lemmas (functoriality)
• Constructor from stochastic processes

Càdlàg paths (LeanLevy/Processes/Cadlag.lean)

• Right-continuity with left limits
• Monotone ℕ-valued functions are càdlàg when right-continuous

Lévy processes (LeanLevy/Processes/LevyProcess.lean)

• Definition: independent increments, stationary increments, càdlàg paths, starts at zero
• Characteristic exponent and supporting lemmas (multiplicativity, non-vanishing, right-continuity)

Poisson process (LeanLevy/Processes/PoissonProcess.lean)

• Definition as a counting process with independent Poisson-distributed increments
• Shown to be a Lévy process

Characteristic exponent (LeanLevy/Levy/CharacteristicExponent.lean)

• Local log construction (branch-cut safe) and local-global exponent agreement
• Semigroup API: multiplicativity, power formulas, complex power law φ_t(ξ) = φ₁(ξ)^t
• Ceiling-sequence density lemma: right-continuous + continuous functions agreeing on ℕ/ℕ rationals are equal
• Lévy exponential formula F(t,ξ) = exp(tΨ(ξ)) with full continuity in t

Infinite divisibility (LeanLevy/Levy/InfiniteDivisible.lean)

• Iterated convolution, with characteristic function formula
• Poisson distribution is infinitely divisible
• Lévy process marginals are infinitely divisible

Lévy measures (LeanLevy/Levy/LevyMeasure.lean)

• IsLevyMeasure predicate: ν({0}) = 0 and ∫ min(1, x²) dν < ∞
• Finite mass on {x | ε ≤ |x|}, σ-finiteness

Compensated integrand (LeanLevy/Levy/CompensatedIntegral.lean)

• levyCompensatedIntegrand ξ x = exp(ixξ) − 1 − ixξ·1_{|x|<1}
• Pointwise norm bound, measurability, Bochner integrability under a Lévy measure

Lévy–Khintchine representation (LeanLevy/Levy/LevyKhintchine.lean)

• LevyKhintchineTriple structure: drift, Gaussian variance, Lévy measure
• Statement of the representation theorem

Incomplete

Two results remain sorry'd:

• exists_poissonProcess — Existence of a Poisson process. Needs the Kolmogorov extension theorem, which isn't in mathlib yet.
• levyKhintchine_representation — The Lévy–Khintchine representation theorem.

Building

Requires Lean 4 (v4.28.0-rc1) and mathlib.

lake build

Structure

LeanLevy/
├── Basic.lean
├── Fourier/
│   └── MeasureFourier.lean
├── Probability/
│   ├── Characteristic.lean
│   ├── Poisson.lean
│   └── WeakConvergence.lean
├── Processes/
│   ├── Cadlag.lean
│   ├── FiniteDimensional.lean
│   ├── ProjectiveFamily.lean
│   ├── Kolmogorov.lean
│   ├── LevyProcess.lean
│   ├── PoissonProcess.lean
│   └── StochasticProcess.lean
└── Levy/
    ├── CharacteristicExponent.lean
    ├── CompensatedIntegral.lean
    ├── InfiniteDivisible.lean
    ├── LevyKhintchine.lean
    └── LevyMeasure.lean