Plain-English statement
The diagonal carre du champ is nonnegative because it is the infinitesimal limit of a nonnegative Markov Jensen gap.
Mathematical statement
If (P_h f(x)) squared is at most P_h(f squared)(x), and the right generator exists on f and f squared, then Gamma(f,f)(x) is nonnegative.
Intuition
A Markov operator cannot increase the square of an average beyond the average square. The infinitesimal version of that variance gap is Gamma.
Conditions
- the pointwise Jensen inequality holds for every positive h
- the right generator limits exist for f and f squared
- P_h f(x) converges to f(x) from the right
Why these conditions cannot be dropped
- Jensen supplies finite-time nonnegativity
- both generator limits identify the derivative of the gap
- orbit continuity turns the difference-of-squares factor into twice f(x)
Proof route
- form the nonnegative Jensen-gap quotient
- rewrite it as a combination of two right difference quotients
- take the product and linear-combination limits
- use closedness of the nonnegative half-line
Lean interface notes
- time is NNReal and the limit filter is nhdsWithin 0 (Ioi 0)
- the proof uses Tendsto arithmetic and isClosed_Ici.mem_of_tendsto
- Gamma positivity is concluded rather than passed as an assumption
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Lean statement
theorem carreDuChamp_nonneg_of_markov_jensen_rightGenerator
(P : ℝ≥0 → (X → ℝ) → X → ℝ)
(generator : (X → ℝ) →ₗ[ℝ] (X → ℝ))
(f : X → ℝ) (x : X)
(hjensen : ∀ h : ℝ≥0, 0 < h → (P h f x) ^ 2 ≤ P h (f * f) x)
(hf : Tendsto
(fun h : ℝ≥0 => (P h f x - f x) / (h : ℝ))
(𝓝[>] 0) (𝓝 (generator f x)))
(hf2 : Tendsto
(fun h : ℝ≥0 => (P h (f * f) x - (f x) ^ 2) / (h : ℝ))
(𝓝[>] 0) (𝓝 (generator (f * f) x)))
(hcontinuous : Tendsto (fun h : ℝ≥0 => P h f x)
(𝓝[>] 0) (𝓝 (f x))) :
0 ≤ carreDuChamp generator f f x := by
let gap : ℝ≥0 → ℝ := fun h =>
(P h (f * f) x - (P h f x) ^ 2) / (2 * (h : ℝ))
have hgap_nonneg : ∀ᶠ h in 𝓝[>] (0 : ℝ≥0), 0 ≤ gap h := by
filter_upwards [self_mem_nhdsWithin] with h hh
have hh0 : 0 < h := by simpa only [mem_Ioi] using hh
exact div_nonneg (sub_nonneg.mpr (hjensen h hh0)) (by positivity)
have hrewrite : ∀ᶠ h in 𝓝[>] (0 : ℝ≥0),
gap h =
(2 : ℝ)⁻¹ * ((P h (f * f) x - (f x) ^ 2) / (h : ℝ)) -
(2 : ℝ)⁻¹ * (((P h f x - f x) / (h : ℝ)) *
(P h f x + f x)) := by
filter_upwards [self_mem_nhdsWithin] with h hh
have hh0 : (h : ℝ) ≠ 0 := by
have : 0 < h := by simpa only [mem_Ioi] using hh
exact_mod_cast this.ne'
dsimp [gap]
field_simp
ring
have hlimit : Tendsto gap (𝓝[>] (0 : ℝ≥0))
(𝓝 (carreDuChamp generator f f x)) := by
have hconst : Tendsto (fun _ : ℝ≥0 => f x)
(nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) :=
tendsto_const_nhds
have hsum : Tendsto (fun h : ℝ≥0 => P h f x + f x)
(nhdsWithin 0 (Set.Ioi 0)) (nhds (f x + f x)) :=
hcontinuous.add hconst
have hprod := hf.mul hsum
have hcombined := (hf2.const_mul (2 : ℝ)⁻¹).sub
(hprod.const_mul (2 : ℝ)⁻¹)
have hvalue :
(2 : ℝ)⁻¹ * generator (f * f) x -
(2 : ℝ)⁻¹ * (generator f x * (f x + f x)) =
carreDuChamp generator f f x := by
simp only [carreDuChamp]
ring
have htarget : Tendsto
(fun h : ℝ≥0 =>
(2 : ℝ)⁻¹ * ((P h (f * f) x - (f x) ^ 2) / (h : ℝ)) -
(2 : ℝ)⁻¹ * (((P h f x - f x) / (h : ℝ)) *
(P h f x + f x)))
(𝓝[>] 0) (𝓝 (carreDuChamp generator f f x)) := by
rw [← hvalue]
exact hcombined
exact htarget.congr' (hrewrite.mono fun h hh => hh.symm)
exact isClosed_Ici.mem_of_tendsto hlimit hgap_nonneg
/-- Chewi Theorem 1.2.14: stationarity and generator symmetry imply the
fundamental integration-by-parts identity between the Dirichlet form and the
integrated carre du champ.
The three integrability hypotheses are the exact terms expanded from Gamma;
they prevent the totalized Bochner integral from hiding a domain failure. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:87published source at 7bcd37294df1