integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport

Plain-English statement

For a twice continuously differentiable compactly supported test function, the exact Gibbs-weighted Langevin generator display is integrable over the whole Euclidean space.

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

If V is C1 and f is C2 with compact support, then exp(-V) (Delta f - <grad V, grad f>) belongs to L1(dx).

Intuition

Outside the compact support of f, local equality with zero forces both its gradient and Laplacian to vanish. The generator display therefore has compact support even when the potential gradient is unbounded.

Conditions

The potential V is globally C1.
The test function f is globally C2.
The test function f has compact support.

Why these conditions cannot be dropped

C1 regularity makes grad V continuous on the compact region where the test function derivatives can be nonzero.
C2 regularity makes grad f and Delta f continuous and gives genuine derivative semantics.
Compact support supplies whole-space domination without imposing a false bounded-gradient condition on a strongly convex potential.

Proof route

Use the ASTIS C1-gradient and C2-Laplacian continuity leaves to prove continuity of the displayed integrand.
At a point outside tsupport f, rewrite f locally as the zero function.
Use fderiv congruence for grad f and Mathlib Laplacian congruence for Delta f.
Conclude that the display is supported inside tsupport f.
Apply continuous integrability for compactly supported functions.

Lean interface notes

HasCompactSupport means compact topological support, not merely bounded pointwise support.
Mathlib derivatives are totalized, so the ContDiff hypotheses retain their analytic meaning.
This theorem is an integrability certificate; the cutoff limit, weighted IBP, generator domain, and invariant law remain separate declarations.

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

theorem integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport
    {n : ℕ}
    {V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    (hV : ContDiff ℝ 1 V)
    (hf : ContDiff ℝ 2 f)
    (hf_support : HasCompactSupport f) :
    Integrable
      (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
        Real.exp (-V y) *
          (Laplacian.laplacian f y - inner ℝ (gradient V y) (gradient f y)))
      volume := by
  have hcontinuous : Continuous
      (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
        Real.exp (-V y) *
          (Laplacian.laplacian f y - inner ℝ (gradient V y) (gradient f y))) := by
    exact hV.continuous.neg.rexp.mul
      ((_root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.continuous_laplacian_of_contDiff_two
          hf).sub
        ((_root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.continuous_gradient_of_contDiff_one
            hV).inner
          (_root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.continuous_gradient_of_contDiff_one
            (hf.of_le (by norm_num)))))
  have hsupport : HasCompactSupport
      (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
        Real.exp (-V y) *
          (Laplacian.laplacian f y - inner ℝ (gradient V y) (gradient f y))) := by
    refine HasCompactSupport.of_support_subset_isCompact hf_support ?_
    intro y hy
    by_contra hy_mem
    have hzero : f =ᶠ[nhds y] (0 : EuclideanSpace ℝ (Fin (n + 1)) → ℝ) := by
      rwa [← notMem_tsupport_iff_eventuallyEq]
    have hgrad : gradient f y = 0 := by
      rw [gradient, hzero.fderiv_eq]
      simp
    have hlap : Laplacian.laplacian f y = 0 := by
      have hzero_lap := InnerProductSpace.laplacian_congr_nhds hzero
      have hzero_lap_y := mem_of_mem_nhds hzero_lap
      calc
        Laplacian.laplacian f y =
            Laplacian.laplacian
              (0 : EuclideanSpace ℝ (Fin (n + 1)) → ℝ) y := hzero_lap_y
        _ = 0 := by
          change Laplacian.laplacian
            (fun _ : EuclideanSpace ℝ (Fin (n + 1)) => (0 : ℝ)) y = 0
          rw [InnerProductSpace.laplacian_const]
          rfl
    exact hy (by simp [hlap, hgrad])
  exact hcontinuous.integrable_of_hasCompactSupport hsupport

/-- Raw finite-coordinate form of
`integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport`.

This transports the Euclidean-space result through Mathlib's
volume-preserving `WithLp.toLp 2` equivalence.  Its conclusion is in the exact
shape consumed by the radial-cutoff dominated-convergence theorem.  It does
not itself take a cutoff limit or prove weighted integration by parts,
generator-domain semantics, stationarity, or invariance. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1017published source at 7bcd37294df1