carreDuChamp · Samplinglib

Plain-English statement

The carre du champ measures the part of the generator applied to a product that is not explained by the ordinary product rule.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

Gamma(f,g) = (L(fg) - f Lg - g Lf)/2.

Intuition

For a diffusion generator, the defect isolates the quadratic gradient term. The abstract definition itself does not assert that this term is nonnegative.

Conditions

L is a real linear map on real-valued observables
products of the selected observables are available in the ambient function space

Why these conditions cannot be dropped

linearity is the generator structure used by later semigroup arguments
positivity requires Markov or diffusion structure and therefore cannot be hidden in this definition

Proof route

define the expression pointwise
derive symmetry separately from commutativity of multiplication

Lean interface notes

the generator is represented by LinearMap
function multiplication is pointwise
the factor one half is represented in the real scalar field

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

How to read the exact Lean declaration

Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.

NameWhat reusable mathematical fact is being created?
ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
PropositionAfter the colon, translate the Lean expression back into a paper statement.
Proof actionsAfter by, ask what each tactic does to the mathematical goal—not only what syntax it uses.

Syntax used on this page

This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.

Source voice and ASTIS voice stay separate

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

def carreDuChamp
    (generator : (X → ℝ) →ₗ[ℝ] (X → ℝ))
    (f g : X → ℝ) : X → ℝ :=
  fun x => (2 : ℝ)⁻¹ *
    (generator (f * g) x - f x * generator g x - g x * generator f x)

/-- The carre du champ is symmetric in its observable arguments. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:35published source at 7bcd37294df1