Plain-English statement
- Finite-box trace integrability for the explicit Gibbs-weighted Langevin field under global `C¹/C²` regularity, with the field derivative chosen as Mathlib's `fderiv`. This removes the remaining supplied `hF` input from `integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff`, but only for the canonical derivative representative. It is still a finite-box regularity handoff, not weighted integration by parts, boundary cancellation, a generator domain theorem, an invariant law, reversibility, or KL/FI dissipation.
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Lean statement
theorem integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff_fderiv
{n : ℕ}
(a b : Fin (n + 1) → ℝ)
(V f : EuclideanSpace ℝ (Fin (n + 1)) → ℝ)
(hV : ContDiff ℝ 1 V)
(hf : ContDiff ℝ 2 f) :
IntegrableOn
(fun x : Fin (n + 1) → ℝ =>
∑ i,
(fderiv ℝ
(fun z : Fin (n + 1) → ℝ => fun i =>
Real.exp (-V (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))) *
fderiv ℝ f (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))
(EuclideanSpace.single i (1 : ℝ))) x)
(Pi.single i (1 : ℝ)) i)
(Set.Icc a b) volume := by
let F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ :=
fun z => fun i =>
Real.exp (-V (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))) *
fderiv ℝ f (WithLp.toLp 2 z : EuclideanSpace ℝ (Fin (n + 1)))
(EuclideanSpace.single i (1 : ℝ))
have hF : ∀ x ∈ Set.Icc a b, HasFDerivAt F (fderiv ℝ F x) x := by
intro x _
simpa [F] using
hasFDerivAt_expNeg_fderivCoordinateField_of_contDiff
(V := V) (f := f) (x := x) hV hf
simpa [F] using
integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff
a b V f (fun x => fderiv ℝ F x) hF hV hf
/-- Whole-space integrability of the concrete Gibbs-weighted Langevin
generator display for a compactly supported `C²` test function.
The compact support belongs to the test function, not to the Gibbs weight.
Outside `tsupport f`, local equality with the zero function forces both
`gradient f` and `Laplacian.laplacian f` to vanish. Thus the full display
`exp (-V) * (Δ f - <∇V, ∇f>)` is continuous and compactly supported even when
`gradient V` is unbounded and the unnormalized Gibbs mass has not been shown
finite.
This is the concrete main-term integrability input for the cutoff route. It
does not prove the cutoff limit itself, weighted integration by parts,
generator-domain semantics, stationarity, an invariant law, reversibility, or
KL/FI dissipation. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:973published source at 7bcd37294df1
Proof architecture
Chewi Ch.1 Langevin root: finite-box trace `IntegrableOn` for the canonical `fderiv` trace of the explicit Gibbs-weighted first-derivative field under global `C¹/C²` regularity
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `have` creates a named intermediate mathematical fact.
- `simpa` closes the goal after a controlled simplification of a typed result.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Totalized `fderiv` values must not be read as a differentiability theorem.
- 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.