chewi_theorem_1_2_21_forward

Plain-English statement

Poincare coercivity turns the variance dissipation identity into exponential variance decay.

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'(t)=-2D(t) and V(t)≤CD(t), then V(t)≤V(0)exp(-2t/C).

Intuition

Poincare says the current variance cannot be large unless the Dirichlet energy is large; the semigroup immediately dissipates at twice that energy.

Conditions

C is positive
the curve has derivative -2 times its dissipation
the Poincare inequality holds at every time

Why these conditions cannot be dropped

positivity of C is needed to divide by C
the source's concrete variance derivative must be supplied separately

Proof route

multiply V≤CD by the nonnegative factor 2/C
obtain (2/C)V≤2D
invoke the generic scaled-dissipation decay theorem

Lean interface notes

the suffix forward is intentional because the source states an equivalence
the concrete reversible semigroup and converse direction are not hidden in the theorem

Lean learning studio · mathematics → formal proof

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Lean statement

theorem chewi_theorem_1_2_21_forward
    {C : ℝ} (hC : 0 < C) (curve : DissipationCurve 2)
    (hPI : ∀ s : ℝ,
      curve.energy s ≤ C * curve.dissipation s)
    {t : ℝ} (ht : 0 ≤ t) :
    curve.energy t ≤
      curve.energy 0 * Real.exp (-(2 / C) * t) := by
  simpa [sub_zero] using
    chewi_theorem_1_2_21_forward_from hC curve hPI ht

/-- Backward scalar direction of Chewi, Theorem 1.2.21. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:221published source at 7bcd37294df1