tendsto_setIntegral_expNeg_norm_ge_comp_toLp_of_lintegral_expNeg_ne_top

Plain-English statement

If the unnormalized Gibbs weight has finite total mass, then its mass outside expanding Euclidean balls tends to zero.

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 integral exp(-V(x)) dx is finite, then integral over {x : R <= norm(x)} of exp(-V(x)) dx tends to zero as R tends to infinity.

Intuition

An integrable density cannot retain positive mass arbitrarily far into the tail. ASTIS states this in the same raw finite-coordinate representation used by the radial cutoffs.

Conditions

The potential V is continuous, so the Gibbs weight is measurable.
The ENNReal integral of exp(-V) is finite.

Why these conditions cannot be dropped

Tail convergence is an L1 consequence and therefore needs finite total mass.
Continuity is stronger than the minimal measurability assumption, but matches the current Chapter 1 Langevin interface and avoids a hidden measurable-representative choice.

Proof route

Convert finite ENNReal Gibbs mass into real-valued Integrable on Euclidean space.
Transport integrability through the volume-preserving PiLp coordinate equivalence.
Apply the generic antitone expanding-ball tail theorem to the Gibbs weight.
Use positivity of Real.exp to remove the scalar norm.

Lean interface notes

The tail family is indexed by real radii and converges in the atTop filter.
The generic tail theorem is stored in Analysis.Calculus.Divergence and is independent of Gibbs measures.
This result does not assemble weighted integration by parts or any semigroup invariance statement.

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Lean statement

theorem tendsto_setIntegral_expNeg_norm_ge_comp_toLp_of_lintegral_expNeg_ne_top
    {n : ℕ}
    {V : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    (hV : Continuous V)
    (hZ : (∫⁻ y : EuclideanSpace ℝ (Fin (n + 1)),
      ENNReal.ofReal (Real.exp (-V y)) ∂volume) ≠ ∞) :
    Tendsto
      (fun R : ℝ => ∫ x in
        {x : Fin (n + 1) → ℝ |
          R ≤ ‖(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))‖},
        Real.exp (-V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
          ∂volume)
      atTop (nhds 0) := by
  have htail :=
    _root_.AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_setIntegral_norm_norm_ge_comp_toLp
      (integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top hV hZ)
  simpa [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] using htail

/-- Whole-space Gibbs-weighted Langevin integration by parts for a compactly
supported `C²` test function:
`∫ exp (-V) * (Δ f - ⟪∇V, ∇f⟫) = 0`.

The proof builds the raw finite-Pi vector field `exp (-V) * Df`, proves that
it is `C¹` and compactly supported, applies the reusable whole-space
coordinate-divergence theorem, identifies its trace with the Langevin
generator display pointwise, and transports volume back to Euclidean space.
No finite Gibbs-mass assumption is needed because the test function is
compactly supported.

This is the analytic core identity used by a later generator-domain and
semigroup-to-invariance bridge.  It does not itself construct a closed
generator, a Markov semigroup, or an invariant probability measure. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1132published source at 7bcd37294df1