File size: 3,107 Bytes
b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | import OAI.Geometry.RCD.Heat.CanonicalStrictPositivity
import OAI.Geometry.RCD.Heat.CanonicalGaussianUpper
import OAI.Geometry.RCD.Heat.BoundedHeatVolumePower
import OAI.Geometry.RCD.Heat.SmallNoiseLogScale
import OAI.Geometry.RCD.Heat.GaussianLogUpper
import OAI.Geometry.RCD.Heat.GaussianActionSlack
namespace OAI
/-! # Uniform sharp upper action on bounded endpoint sets -/
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped NNReal ENNReal Topology
noncomputable section
namespace RegularPointCoordinates.NoncollapsedRCD
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X)
(hn : 2 ≤ n) (p : X) [SeparableSpace X]
[IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular]
theorem canonicalHeatKernel_bounded_log_upper
(ε : ℕ → ℝ) (hε : ∀ k, 0 < ε k) (hε1 : ∀ k, ε k ≤ 1)
(hε0 : Tendsto ε atTop (𝓝 0)) {R T η : ℝ}
(hR : 0 ≤ R) (hT : 0 < T) (hT1 : T < 1) (hη : 0 < η) :
∀ᶠ k in atTop, ∀ x ∈ closedBall p R, ∀ y ∈ closedBall p R,
ε k*Real.log (hX.canonicalHeatKernel hn p (ε k*T/2) x y) ≤ -(dist x y)^2/(2*T)+η := by
let a := η*T/(2*(R^2+1))
have ha : 0 < a := by dsimp only [a]; positivity
obtain ⟨C,D,hC,_hD,hupper⟩ := hX.canonicalHeatKernel_gaussian_upper hn p ha
obtain ⟨c,hc,hvolume⟩ := hX.bounded_heat_volume_power hn p R
let E := fun k => ε k*Real.log (C/c) - (n:ℝ)*ε k/2*Real.log (ε k*T/2) + D*(ε k)^2*T/2
have hlog : Tendsto (fun k => ε k*Real.log (ε k*T/2)) atTop (𝓝 0) := by
simpa only [mul_div_assoc] using small_noise_log_scale hε hε0 (half_pos hT)
have hElim : Tendsto E atTop (𝓝 0) := by
have hh := ((hε0.mul_const (Real.log (C/c))).sub (hlog.const_mul ((n:ℝ)/2))).add
(((hε0.pow 2).const_mul D).mul_const (T/2))
convert hh using 1
· funext k; dsimp only [E]; ring
· norm_num
filter_upwards [hElim.eventually (gt_mem_nhds (half_pos hη))] with k hk
intro x hx y hy
have ht : 0 < ε k*T/2 := div_pos (mul_pos (hε k) hT) (by norm_num)
have ht1 : ε k*T/2 ≤ 1 := by
have hh := mul_le_mul_of_nonneg_right (hε1 k) hT.le
nlinarith
have hp := hX.canonicalHeatKernel_pos hn p (ε k*T/2) ht x y
have hvol := hvolume y hy (ε k*T/2) ht ht1
have hh := gaussian_log_upper hp hC hc ht (hε k).le hvol (hupper (ε k*T/2) ht x y)
have he : ε k*(-(dist x y)^2/((4+a)*(ε k*T/2))+D*(ε k*T/2)) =
-2*(dist x y)^2/((4+a)*T)+D*(ε k)^2*T/2 := by
field_simp [ne_of_gt (hε k),ne_of_gt hT,ne_of_gt (by linarith : 0 < 4+a)]
rw [he] at hh
simp only [neg_mul,neg_div] at hh
have hd : dist x y ≤ 2*R := by
have hx' : dist x p ≤ R := hx
have hy' : dist y p ≤ R := hy
have hd := dist_triangle x p y
rw [dist_comm p y] at hd
linarith
have hs := gaussian_action_slack hT hη hR (dist_nonneg (x := x) (y := y)) hd
change (dist x y)^2/(2*T)-2*(dist x y)^2/((4+a)*T) ≤ η/4 at hs
dsimp only [E] at hk
rw [neg_div]
nlinarith
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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