scaled_dissipation_of_exponential_decay

Plain-English statement

Exponential decay after every starting time forces the instantaneous coercivity inequality at each starting point.

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 E(s+t)≤E(s)exp(-rt) for every s and t≥0, and E'(s)=-aD(s), then rE(s)≤aD(s).

Intuition

At time s, the true energy curve lies below its tangent exponential envelope and touches it. Their difference has a one-sided local maximum, so its right derivative cannot be positive.

Conditions

the exact right derivative E'(s)=-aD(s) exists
the decay estimate is valid after every starting time, not only from time zero
the comparison exponential uses the same rate r

Why these conditions cannot be dropped

decay only from time zero does not determine the derivative at a later arbitrary time
the one-sided derivative and positive tangent direction match nonnegative semigroup time

Proof route

subtract the shifted exponential envelope from the energy
show the difference has a local maximum at the starting time on [s,∞)
apply one-sided Fermat in the positive tangent direction
rewrite the derivative inequality as coercivity

Lean interface notes

the positive direction 1 is proved to lie in the tangent cone using the nhdsGT filter
the theorem is scalar and does not infer a density or generator domain

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Lean statement

theorem scaled_dissipation_of_exponential_decay
    {scale rate : ℝ} (curve : DissipationCurve scale)
    (hdecay : ∀ s t : ℝ, 0 ≤ t →
      curve.energy (s + t) ≤
        curve.energy s * Real.exp (-rate * t)) :
    ∀ s : ℝ,
      rate * curve.energy s ≤ scale * curve.dissipation s := by
  intro s
  let comparison : ℝ → ℝ := fun x =>
    curve.energy x -
      curve.energy s * Real.exp (-rate * (x - s))
  have hmax : IsLocalMaxOn comparison (Ici s) s := by
    filter_upwards [self_mem_nhdsWithin] with x hx
    have hdecay_x :
        curve.energy x ≤
          curve.energy s * Real.exp (-rate * (x - s)) := by
      have h := hdecay s (x - s) (sub_nonneg.mpr hx)
      calc
        curve.energy x = curve.energy (s + (x - s)) := by
          congr 1
          ring
        _ ≤ curve.energy s * Real.exp (-rate * (x - s)) := h
    dsimp [comparison]
    calc
      curve.energy x -
          curve.energy s * Real.exp (-rate * (x - s)) ≤ 0 :=
        sub_nonpos.mpr hdecay_x
      _ = curve.energy s -
          curve.energy s * Real.exp (-rate * (s - s)) := by simp
  have hinner :
      HasDerivAt (fun x : ℝ => -rate * (x - s)) (-rate) s := by
    change HasDerivAt (fun y : ℝ => -rate * (id y - s)) (-rate) s
    have hraw := ((hasDerivAt_id s).sub_const s).const_mul (-rate)
    exact hraw.congr_deriv (by ring)
  have hexponential :
      HasDerivAt
        (fun x : ℝ =>
          curve.energy s * Real.exp (-rate * (x - s)))
        (-rate * curve.energy s) s := by
    have hraw := hinner.exp.const_mul (curve.energy s)
    apply hraw.congr_deriv
    rw [sub_self, mul_zero, Real.exp_zero]
    ring
  have hderiv :
      HasDerivWithinAt comparison
        ((-scale * curve.dissipation s) -
          (-rate * curve.energy s))
        (Ici s) s := by
    change HasDerivWithinAt
      (curve.energy - fun x : ℝ =>
        curve.energy s * Real.exp (-rate * (x - s)))
      ((-scale * curve.dissipation s) - (-rate * curve.energy s))
      (Ici s) s
    exact (curve.energy_hasDerivWithinAt s).sub
      hexponential.hasDerivWithinAt
  have hone : (1 : ℝ) ∈ posTangentConeAt (Ici s) s := by
    rw [one_mem_posTangentConeAt_iff_frequently]
    have hev : ∀ᶠ x in 𝓝[>] s, x ∈ Ici s := by
      filter_upwards [self_mem_nhdsWithin] with x hx
      have hsx : s < x := by
        simpa only [mem_Ioi] using hx
      exact hsx.le
    exact hev.frequently
  have hnonpos :=
    hmax.hasFDerivWithinAt_nonpos hderiv.hasFDerivWithinAt hone
  have hscalar :
      ((-scale * curve.dissipation s) -
        (-rate * curve.energy s)) ≤ 0 := by
    simpa using hnonpos
  linarith

/-- Forward direction of Chewi, Theorem 1.2.21, between arbitrary times. -/

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:130published source at 7bcd37294df1