integrable_expNeg_langevinGenerator_rhs_comp_toLp_of_contDiff_of_hasCompactSupport

Plain-English statement

- 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.

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.

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Lean statement

theorem integrable_expNeg_langevinGenerator_rhs_comp_toLp_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 x : Fin (n + 1) → ℝ =>
        Real.exp (-V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) *
          (Laplacian.laplacian f
              (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) -
            inner ℝ
              (gradient V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
              (gradient f (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))))
      volume := by
  have hintegrable :=
    integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport
      (V := V) (f := f) hV hf hf_support
  rw [← (PiLp.volume_preserving_toLp (Fin (n + 1))).integrable_comp_emb
    (MeasurableEquiv.toLp 2 _).measurableEmbedding] at hintegrable
  simpa [Function.comp_def] using hintegrable

/-- Whole-space integrability of the unnormalized Gibbs weight in raw
finite-Pi coordinates.

The hypothesis is the finite `ℝ≥0∞` Gibbs mass on Euclidean space.  Continuity
supplies measurability, and Mathlib's volume-preserving `WithLp.toLp 2`
equivalence transports integrability to the coordinate representation used by
the cutoff lemmas.  No test function, generator, tail limit, IBP, or invariant
law is asserted here. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1074published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Example 1.2.8 cutoff route: expose concrete generator-display integrability in the raw finite-Pi coordinate shape consumed by radial-cutoff dominated convergence

Lean proof walkthrough

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Why the statement has this shape

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Pointwise support, topological support, and compact support retain distinct meanings.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.